IIRC, [a long time ago?] you MUST give a piece of paper to a guy/gal near the exit, so they remember to mark your exit in the database. I don't think they check too much, just collect the pieces of papers? (I don't remember the details, but if there was a scary warning if you forget to give it and they forget to update the database.)
This is part of the Canadian border control in my experience; i.e.
- Arrive in Pearson
- go to passport control, scan passport at kiosk, kiosk spits out receipt
- you flash receipt to a border officer at the stairs out of passport control
- go to baggage claim
- get bags
- go to customs
- hand passport with receipt to officer at customs
- welcome to Canada
I haven't had that experience when leaving other countries bound for the US; like GP said, I've had to show my passport to the airline to board, but I believe that's to fulfill the requirement that they made good-faith effort to ensure I'm capable of entering the US.