At-home sleep studies measure heart rate (maybe ekg, some probe is on your skin), blood oxygen, sound, and the direction of gravity relative to your testing device (to classify results by "back", "side", "prone", etc.)
I did one of these and it was the worst night of sleep I ever had. It's somewhat invasive. Meanwhile, I sleep with my watch every night with no problems.
Amusingly my Apple Watch pulse ox showed low so I did a sleep study. It's invasive, but it plugs into your nostrils (and in my case held them open). Slept like a baby. Next day 99% was the lowest reading. No problem. Now I use these sleep strips from this company called Intake. Sleep well. Holds nose open. I have a deviated septum, apparently. I think it's from falling off a tree onto my nose as a child, and being hit in the face by basketball numerous times. But probably not.
What was funny was that at-home sleep study was damned good sleep because it held my nose open!
I think this depends on your doctor/clinic and where you are. The last one I took home just had an oxygen sensor and the skin probe/patch. The doctor said they get (hand wavey) 95%+ of the same diagnostic capability from the simpler setup and far fewer claims of poor quality sleep caused by the equipment.
They really clamp your finger in the pulse oximeter probe, and it's taped with one-time-use adhesive. So there is no taking it out when it hurts, you just have to live with it and try to sleep. The instructions say "it's not that bad, you'll be fine".
(My flimsy understanding is that blood oxygen measurement is how at-home sleep studies work?)