When flying from a Schengen area country to the US, border patrol on the departing side is entirely managed by that country. Often e.g. United-operated flights from larger European airports will have their own passport check, but that always happens after you pass through the departing country's passport control. At least, this is my personal experience flying between Germany and the US ~once a month.

I have been to a few airports where you explicitly clear US customs at the departing airport before leaving for the US — Vancouver, BC and Dublin Shannon come to mind.

If you've overstayed your automatic 90-day Schengen area travel visa, a common hack for backpackers and such is to intentionally leave the Schengen area from someplace like France that's generally less strict about counting exact days than many other EU countries.

To be fair, I don't have much experience flying from the Schengen area to non-Schengen area non-US countries. It's totally possible this is all rigamarole that the US government pressures other countries to do.

Over the past few years, I've flown in and out of the EU dozens of times. Every EU international airport I've flown out of has outbound passport control, operated by the local country, before you enter the international part of the terminal.

I fly between Germany and UK pretty frequently, and whilst the UK doesn't check the passport on the way out, Germany do check it on the way out (you get checked again when entering into UK).

I have heard of friends of friends who overstayed a visa and get found out on the exit gate (usually bringing a lawyer/embassy official to talk through things). This can lead to a (temporary?) travel ban to Germany and a fine.