Can a rogue radio signal from the moon or near orbit realistically go unnoticed?

Hide it in plain sight? Tell everyone you are putting some science experiments in the moon. Then no one will surprised when you start transmitting large amounts of data from the Moon back to the Earth. Some of the data could be actual science as a cover, while you encode your secret information as extra data in the broadcast.

I did not see it mentioned anywhere else, so I thought I would add: If you use a parabolic or other type of beam-focusing antenna, it's possible that only someone at or near your receiver location could even detect the signal. Combined with the other obfuscation techniques mentioned in other answers, this could further help conceal your signal.

There is a good chance they will be noticed

Is anyone watching the moon closely?

Thanks to Donald Trump's Space Policy Directive–1 (AKA: "the Boots on the Moon directive"), the answer to this is almost certainly yes. According to this, it is the USA's goal for the "return of humans to the Moon for long-term exploration and utilization". This means setting up a base on the moon. Being that this directive was sent to the secretaries of Defense and Homeland Security, this implies the involvement of military/strategic interest in this directive.

If this policy is considered to include the pursuit of military objectives, then they need to closely monitor the moon while this mission is being planned to keep an eye on any foreign powers who attempt to set up bases of their own which might interfere with American interests and vise-versa. As you might expect, military surveillance is quite a bit more in depth than casual civilian observations because they will be looking for what may be intentionally hidden.

Can you tell the difference between background noise, a moon bounce, and a moon transmission?

Yes. The key difference between a reflected signal and a transmitted signal is how it triangulates. A reflected signal will bounce off of a very large surface area meaning that if you were to use multiple receivers on Earth to triangulate it's point of origin, it would have an apparent point of origin somewhere behind the moon. Background noise will triangulate WAY farther back since it will mostly be originating from other solar systems. In contrast, a transmission coming from somewhere on or in orbit of the moon will have an apparent point of origin somewhere that you have line-of-sight indicating that it is actually coming from a point and not just a broad area reflection. A military organization interested in monitoring signals from the moon could use multiple receivers to filter out all the noise cause by reflected and background signals and isolate just signals transmitted from the actual moon.

How this works is that you have multiple receivers that will pick up 3 slightly different noise patterns. From any one vantage point it is just random static but from 3 vantage points you will see the same noise patterns repeated, but with different displacements based on apparent distance. By overlapping the images based on an expected focal distance you will get triple images for all noise patterns that are not at that distance; so, Background noise will be the most displaced, moon bounced signals will be slightly displaced, and signals originating from the moon will line perfectly leaving you just the exact position of the signal's origen.

Can you hide this signal?

For the most part, yes... point-to-point transmitters like laser transmitters and Directional Antennas can send a directed signal between two points without radiating out in every direction for ground based receivers to intercept, but these technologies can be unreliable when used at astronomical distances because minor disturbances can cause you to lose track of the object you are trying to communicate with. Any spacecraft that is expected to lose line-of-sight at any point in its mission is typically outfitted with an omni-directional LGA or wide area MGA transmitter in addition to its HGA point-to-point transmitter to help maintain and re-establish lost signals. So, what your aliens are communicating with on Earth will matter a lot. If they are communicating between two fixed position communication points then they only need an HGA and they can stay hidden. However, if they are trying to communicate with a mobile exploration team on the Earth's surface, or if they are trying to establish communications with a variety of receivers, then they will likely need something in the wider beam width range. This would risk their detection.

Will you it be detected anyway?

Probably. A military level interest in the moon means that it's not just being listened to, but watched. Modern telescopes, IR sensors, radar sweeps, and lidar sweeps make it relatively easy to map out the surface of the moon, and anything that might be in orbit of it. This is where modern technology really comes into play. A few decades ago, you could have hidden on the moon, just because it is big enough and the labor needed to analyze the data of the whole moon's surface would have been cost prohibitive, but modern AI pattern recognition and stealth detection techniques have become adequate that it would be very difficult for a communications station to go unnoticed unless it was constructed and effectively obfuscated years ago before we had the ability to keep an eye on things up there.

This is related to @JohanHagström's suggestion but somewhat "harder". I felt it would be inappropriate to do a "delete all and insert" on his answer.

SETI, and the related question of how much recognisable information we're broadcasting to the Universe, and the related SF trope of approaching aliens watching our TV, is predicated on the fact that during C20th all of the morse, audio and (later) video that we transmitted had either no or some extremely simple encoding which would make it trivial to decode (for one treatment, see http://www.fourmilab.ch/documents/sftriple/gpic.html)

However over the last couple of decades there has been a tendency to replace traditional data/audio/video transmissions with more complex encodings, overlayed on a pseudo-random sequence of "chips" (see https://en.wikipedia.org/wiki/Chip_(CDMA) and in particular the examples in the first para). Variants of this technique are used by GSM 'phones, direct broadcasting of various kinds, GPS satellites and many other sources of RF both terrestrial and orbital.

If the sequence is long enough, then even if an eavesdropper chances on the signal, it will look like random noise (i.e. "the white noise of the universe" mentioned by Johan) unless he knows the mathematical parameters underlying the pseudo-random sequence. If he hasn't realised that a particular frequency is of interest, then the best he can do is scan all frequencies (or at least, all frequencies within a certain band e.g. the 1.42-67GHz "water hole" band which in principle is kept "quiet" for radio astronomy).

The reason that things like WiFi and GPS work is that the receivers know the transmission parameters, and at the most they have to look for a comparatively small number of predefined sequences. A signal with unknown parameters is difficult to detect unless notably strong, and even if detected difficult to decode with sufficient certainty that it can be flagged as worthy of resources for further study.

Yes, by hiding it plain sight - A signal hidden with encryption in the white noise of the universe would very likely go unnoticed.

On old TV:s you can actually see the white noise of the universe and by adding on tiny seemingly randomized well timed encrypted data bytes it would very likely with our current technology be quite unnoticed.

Meteor Burst Communications 'might' fit the bill.

Wikipedia link

The system would be limited in the amount of data that could be sent with each 'burst' transmission (not so much if its Aliens I guess) but even if a suspicious signal was intercepted (and you'd probably have to be looking for it anyway) detecting the point of origin for any particular transmission would be hellishly difficult. And since modern electronics and phased array antennae can be reasonably compact/transportable it might be pointless to try anyway anyway.

We see signals coming from the Moon all the time because EME has been a thing since after WW2.

Every other amateur radio operator knows how to do it and it's very popular. Do check the wiki linked above. The TL;DR for the article is: lots of people use the Moon as a relay for radio signals on purpose and for fun. Those geeks have even created a festivity of their own, called World Moon Bounce Day:

World Moon Bounce Day will be June 27 in Australia and June 26 in the U.S.(...) Participants worldwide will celebrate the 40th anniversary of the 1969 Apollo 11 moon landing by bouncing radio voice transmissions off the moon, known as "moon bounce."

In collaboration with schools, amateur radio organizations, and cultural groups, people from around the world will communicate with one another via the moon (...) Station operators and their guests will experience the thrill of hearing their own voice or that of others talking via the moon (...)

If your aliens' signal is coming from anywhere close to the Moon's surface, it will be noticed but it will be just a droplet in an ocean of signals. This is the best kind of stealth, as everybody will automatically assume it's just a couple of DIY dorks ham radio enthusiasts playing with their ham stations and won't ever bother to peek. If the aliens can use human encryption with a strong key then they will really be safe as it will look even more like users from diy.se ham.stackexchange.com and retrocomputing.se at play.

For all we know, the aliens could be doing that right now and we wouldn't be able to tell. I would say this jokingly if it were not 2020.

Ash might be on to something, about the moon. Nobody listens to the moon, because it reflects garbage noise signals back from earth.

How does one tell if the signal originated from the moon, or was just a bounce back from earth? And who would care?

So use the moon to screen your satelite from Earth. It would take a very powerful engine to give it enough velocity to keep up with the moon, seeing as its orbit around earth would be further away from Earth than the moon, but that is not I presume within the scope of the question.

So your satelite just peeks out from behind the moon, to give LoS with the Earth. Maybe even use gravitational lensing, but that is a stretch. Any signal that is accidentally detected from earth would look like just another uninteresting artifact, random noise, some earth transmitter sending funny signals into space and bouncing off the moon, a wayward reflection from a satelite signal from some rogue earth satelite gone bonkers. A curiosity at best, but nothing of significance. Nothing to report.

Who woulld possibly look for something in the corona of the moon? And what would it even look like? A dot on a mountain in profile?

It would take multiple receivers on earth, all in sync, listening to exactly the same signal, to identify it as not just a bounce-back of a signal from some weirdos on earth. You would have to be intentionally looking for it, knowing exactly what you were looking for.

But do it quickly, the back side of the moon is starting to get crowded, and the orbits around the moon are getting as cluttered with junk as the space around Earth.

Yes, if it's cleverly designed:

Depends on signal to noise ratio

If it's just putting out a "beep", like Sputnik 1, it will attract attention. That's clearly an artificial signal, it stands out to anyone casually looking at the frequency spectrum, its easily detected, and it will attract some investigation.

The more random the signal looks, the less likely it is to attract attention. If there are repeating features in the message, then it will be easy to detect. If your signal looks like white noise, then it will be a lot harder to detect.

The best way to get white noise, compress or encrypt the data, have no or tiny fixed packet structure to give landmarks to anyone observing, use spread spectrum, and send it at the highest speed the lowest power setting your transmitter will allow.

Also, I don't think anyone is listening to the moon.

SETI are pointing radio telescopes at interesting places in deep space. As far as I'm aware, they're not pointing the radio telescope at the moon, and for good reason, it reflects our own signals back to us.

Ham radio operators will still use the moon for signal relay, but to them white noise will sound like white noise. It will be ignored.

Nobody from the commercial world will investigate where a rogue signal is coming from unless it affects their transmission quality. If the builders did some research on spectrum allocation, they could find a part of the frequency spectrum that isn't being actively used by anyone else (so wont cause anyone else to go investigate why their uplink suddenly died).

Unnoticed doesn't mean undetected. It can also mean people detect you, but don't care. There are plenty of satellites in orbit transmitting to various people on the ground and to other satellites.

Establish a shell corporation allegedly working in a field that would be expected to launch satellites: telecommunications is a classic, of course; earth imagery might be another useful field. Do some work to make your cover realistic, and nobody will pay any real attention to your one specific transmitter out of hundreds.

If that's too much work, and you have another way to arrange a launch (or conceal whatever you're using to put the satellite in orbit), you can just have your "communications satellite" appear one day. Someone will presumably wonder who owns it at some point, but I would imagine they'd start investigating possibilities like missing registry paperwork or a provider who's gone out of business before they get to possibilities like secret government black ops, let alone aliens.

TL;DR: anybody can resolve your transmitter with a halfway decent telescope (you require a LoS to Earth, which means Earth has a LoS back) so your best bet is to be dismissed as yet another boring practical satellite.