Generally most sleep tracking products use actigraphy (measurements of your movement) to track sleep. Pebble partnered with researchers at Stanford back in the day to produce an algorithm for our operating system, which worked quite well and is now open source!
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.
It's been a long time since I used my Pebble Time Steel, but I remember the alarm feature to wake you up when you were in a light phase of sleep worked very well. I didn't do any sleep tracking personally, but the watch seemed to be able to tell.
From an old Kickstarter:
"Pebble Health tracks when you fall asleep, wake up, and how much deep sleep you’re getting (that’s the really good stuff). Smart Alarms determine your optimal wake-up time based on your sleep cycle, so you’re less groggy and more energized to tackle the day."
