I own the Pebble 2 Duo and the answer is "almost none". It basically just tells you how long you were still/not moving much in the evening, as you'd expect. It's a pretty good proxy for what time you went to bed and got up, and that's about it. It can't actually tell if you're asleep.

There are also (currently) no sleep metrics on app itself; you can only see them on the watch, which doesn't show much besides the sleep duration and an abstract representation showing where you might have woken up in the night.