Without access to breaker box, how do I determine if outlets are on the same circuit?

If you check both Hots with a meter and they read 240v instead of 120v they are on seperate legs but still may be on the same breaker. This is called a mwbc multiwire branch circuit. Your shed should have GFCI protected circuits.

What you shouldn't do is short one to see if the other breaks too.

First, measure the voltage between the lives of both receptacles. Use an extension cord to get from one close to the other. If it's 240V, then you have two circuits. If it's around 0V you don't know (yet). The extension chord helps you measure between two lives of two different receptacles if the multi-meter probe is not long enough. Remember, you're measuring live-to-live, not live-to-neutral. The live pin of of the socket is the smaller one intended for the smaller plug blade.

Failing insight from the frist test (you got 0V in the above test), do this test: measure the voltage in one receptacle (let's call it "A"). Now you are measuring live-to-neutral in "A". Take note of the reading, it should be about 120V.

Then plug in a large load like a space heater, hair dryer, kettle, AC or mining rig etc... in that same receptacle "A", and measure the voltage. This is a bit tricky, and you can use a power bar to plug in a device and measure at the same time. Power bars are usually not for big loads as intended here, but for a short test this should be fine.

Take note of the reading and observe the voltage drop from 120V in "A". Perhaps you read 118V, corresponding to a 2V drop. The drop could be in the range 1V..3V, and depends on how far you are from the panel and how big the load is.

Then measure the voltage in the other receptacle, call it "B", which is the other one in the room and that you are hoping is on a different circuit. Keep the load running on "A". So now you are loading "A" and measuring live-to-neutral in "B"

If "B" has a similar drop, then both receptacles are on the same circuit. If the drop is notably less (perhaps even 0V), then they are on different circuits.

This trick works if the second circuit for "B" is only lightly loaded. If you can control it, switch of all other heavy loads on "B". Should that not be possible, you can perform this same in opposite order: measure and load "B", and then measure "A" and compare.

Also, it's not laziness that puts two outlets on different circuits. It's material and labour cost, plus it's a desirable feature as you now discover. You have more wattage available in one room, and if one circuit goes out you still have power.