i am looking for a wearable. maybe a watch, a wrist band or a ring.
what is the most privacy respecting way, with preferably free and open source software, of monitoring my sleep?
The best way I know of is finding a device that's supported by gadgetbridge: https://codeberg.org/Freeyourgadget/Gadgetbridge#supported-devices (be sure to check the warnings, some devices require you to pair with the OEM app first or possibly even keep the OEM app around)
Though, I used this with an Amazfit Bip and didn't find the data particularity insightful. It often was just straight up wrong about whether I was sleeping or not. So, I too am curious to see what other answers come in.
I have a PineTime and a Xiaomi Smart Band 7. There has been little work done on sleep tracking for the PineTime using InfiniTime.
I'm tracking my sleep using the Smart Band 7 and Gadgetbridge not Xiaomi's app. You do have to use Xiaomi's app to set up the band the first time, but that's been the case since the Mi Band 4.
Xiaomi's bands combined with FOSS Gadgetbridge is an excellent combination.
If you install wasp os on the pinetime and the sleep tracker linked in one of my other comments on this thread on wasp os, then that should work, but it is a bit of work. Hopefully it will be packaged with the next wasp os.
After going with Xiaomi devices for some years (with Gadgetbridge) I revived my trusty Pebble Time Steel only yesterday. This thing is 8 years old and doesn't do heartrate but it still tracks my sleep more precise than any of Xiaomi's devices ever did. Maybe you can find a device in good condition on eBay? Be sure to ask sellers for daylight pictures of the screen with settings menu or a bright watchface. If the battery is dying the screen looks like nacre cause it's being lifted by the battery.
I have a really basic fitbit with phone notifications turned off, and no auto-wakeup by wrist turn, and it lasts almost 2 weeks on a charge. It's amazing.
As far as privacy goes though, I'm sure it leaves a lot to be desired.