Back in the 1950s my parents employed "Au-Pair" to help look after me. They were mainly from Switzerland. I seem to remember we had to take them to our local police station when they first arrived. Beyond that I'm not sure.
These days, despite there being no obvious check on departures I believe the transport operators (not just the airlines) submit passenger lists electronically (they do it inbound too). The problem is that although that might tell us who has oevrstayed it gives us no clue as to where they are.