They know - it is the case that the Thing takes the form of the victim, not that it makes a copy.

The original Thing was based upon a book called 'Who Goes There?' by John W. Campbell. Great story, look it up!

There are three movie adaptions...

1. The Thing From Another Planet ('59?)
2. The Thing by John Carpenter ('82)
3. And the 2011 Thing.

The John Carpenter version is pretty good adaption, considering the novella was written in the late 30's.

It's mostly suggested that the reason that the alien does this as this is part of its dietary process; the ultimate parasitic carnivore, if you will. Perhaps this process is to fuel its biological processes, and was suggested that the one that landed in Antarctica had escaped a previous planet because everything had been consumed, and had stolen a craft in search of more food. It mimics its prey to be a much more proficient hunter; so pretty much that's the answer.

Obviously, this seems a bit of a quirk in the evolutionary sense. Much like the Bioraptors of Pitch Black, there is no plausible way a terrestrial planet could support such a carnivorous being.

I'm going to go with 'they know' considering that each cell is a nation and that jazz. It would seem weird that you were eaten by a cannibalistic alien, turned into one, and then mindwiped. I'm leaning towards the infiltration route; it's just a dang good liar and self-preserver. You would think the gaps in memory, missing time, and the fact that your blood can think on its own would give it away just a little bit.

Note: I am basing my answer off of the "original" 1982 version ("original" in quotes because I understand from Jersey's answer that there was a 1959 movie based off of the same original book).

Creatures duplicated by the Thing are the Thing. Any behavior that mimics that of the original version of the form they've taken is merely defensive camouflage.

When the Thing got Bennings, for example, it was pretty clear that it was trying to pretend to be Bennings, as opposed to thinking it was Bennings.

Garry: My God, what was happening to him?

MacReady: If it had time to finish, it would have looked and sounded and acted just like Bennings!

Garry: I don't know what you're saying.

MacReady: That was one of those things out there Garry, trying to imitate him. C'mon.

Garry: MacReady, I know Bennings, I've known him for ten years. He's my friend.

MacReady: We've gotta burn the rest of them.

Also, Dr. Blair explains the fundamental process of assimilation pretty clearly:

Dr. Blair: You see, what we're talking about here is an organism that imitates other life forms and it imitates them perfectly. When this thing attacked our dogs it tried to digest them... absorb them. And in the process shape its own cells to imitate them. This for instance. That's not dog. It's an imitation. We got to it before it had time to finish.

So its not a dog, or a copy of a dog that thinks its a dog but can be controlled by the Thing. It's just the Thing in dog form, right down to the cellular level: the cells are Thing cells, shaped like dog cells.

Also note the basic premise of the "blood test" they performed was that every piece of the thing was that every piece of the alien is an individual organism with its own survival instinct that will react defensively when threatened.

Since the "blood test" worked exactly as MacReady predicted, it strongly implies that every cell of the Thing is always... the Thing.

One reason that film is so deeply terrifying is that we never know what really goes on in the minds of those who have been "taken over". In the blood test scene, the men tied to the couches are relieved when their neighbors pass the test and prove to be human, but some of them are even more deeply and visibly relieved when they themselves do.

So what's going on inside? We know three things about a man who has been taken over:

1. He can talk and act just like the man did before; the personality and memory are intact.
2. He (it?) can carry out complex plans requiring alien knowledge and motives.
3. There is no outward communication and no visible conflict. He never says "I come in peace" or "you cannot defeat me!" (and think how lame it would be if he did). He acts suspicious and fearful, just as the untouched man would, even if the original man was not a good liar.

From the first fact, it's hard to avoid the conclusion that the human mind is still present and conscious. (The brain may be made of converted cells, but if they behave the same as the unconverted cells, then the thoughts will be the same-- let's not start arguing about the Chinese Room.) From the second, we know that the alien is in there too, and from the third we conclude that it's in charge.

So how could he not know? What does the human mind think while the body is carrying out sinister plans like destroying the blood bank, or planting false clues? How can one not remember doing such things? Or being attacked in the first place? With access to the brain, the Thing could repress memories, or disrupt them before they become permanent. The man might be unconscious during those intervals, or sleepwalking, or daydreaming, or vaguely believing that he's doing something else.

Nothing they could do would really prove they don't know. And proving they do know would almost certainly weaken the story. So let's enjoy the uncertainty.

The scene in which one character watches the head sprout spidery legs and walk across the floor and the character exclaims something like "You're fucking kidding me!" is interesting because that character is at least already infected. The Thing might be so intelligent that it could feign surprise or there might be a stage at which the infected don't know or The Thing component might always be there, using the old personality when it needs to but able to take over instantly. So in some sense, the infected human never knows -- it is kept separate from the mind of The Thing and even is able to, when The Thing allows it to, act like it was still human and believe it is still human. There may later come a stage in which the original mind is completely gone, also.

I think it highly plausible that a being evolved to assimilate an organism in this way would retain or mimic the consciousness of that organism in a way that wasn't apparent to the consciousness. This would be highly advantageous to its ability to effectively camoflage itself as it stalked additional victims. The "Thing" would operate more in the subconscious, influencing the organism to inconspicuously behave in its true reproductive interest. The outward presenting consciousness would simply be influenced to ignore, forget, or create stories to explain its actions, otherwise oblivious to the true motive of the organism. The conscious "awareness" simply isn't necessary, and would probably be an evolutionary liability to the organism on the whole. Alternatively an organism that believes its legitimate will never act otherwise.

I say this as our brains already work this way to a great extent so it would make sense for the organism to simply hijack it. We percieve the decisions we make and the thoughts we experience as creations of our conscious free will. However, brain research strongly suggests that most of our behaviour is actually controlled subconsciously. The conscious mind we experience is like a rider on an elephant. The elephant is mostly controlling the show. The conscious mind simply has the illusion of control, charged with rationalizing and explaining the behavior of the organism but not in charge of it. (look up split brain experiments, also The Happiness Hypothesis by Jonathan Haidt)

How do you know your blood cells aren't currently conscious? Have you ever tried taking your own blood and poking it with a hot wire? The way that Blair is infected isn't from being eaten and replicated, he's just bitten by the dog, the cells from the dog in the bite wound slowly take over his own cells, replicating and replacing them cell by cell to appear to the rest of his body as exact duplicates, ensuring he doesn't know, you would not even feel this if it went slowly enough, it would be exactly the same as normal cell mitosis and apoptosis.

Each cell in your body does act of it's own accord, most of them without any input from you consciously, if taken over by another intelligence, especially one which does not want to be noticed, leaving the mind intact until you can either infect all your uninfected comrades at once, or are detected by your uninfected comrades, is the best option for blending in and thus infecting more people. Therefore I think the victims don't consciously know they're infected, but the thing controlling them does, their consciousness is intact until the moment it is no longer convenient for the thing that infected them to maintain the charade.