All EU citizens (including Irish) and British citizens show a passport or national identity card when entering Britain or Ireland, or when entering the Schengen area.

The document is checked, but it is not stamped. GB/IE←→Schengen checks used to be very simple — often just a glance, no checking the chip or swiping the passport in a reader. Often it was just one or two customs people standing by the gate and looking at each document, not even with a desk. Now (since the refugee thing?) they do swipe the document on the passport-reader-machine.

I assume an electronic record is kept (at least for some period of time), and it is surely shared within the EU+GB, but I don't know if it would be shared to the USA.

(If the situation were reversed, would the USA tell the EU about American citizens who've crossed the US/Canada border using their driving license?)