I travel in and out of Schengen many times per year. I rarely have stamps in both directions and often in neither. if you depart the area over land you will almost certainly not have a stamp and if you enter by land the same is usually true. That is as long as you have a passport from a visa waiver country, as the US is.

They do not check passport stamps to verify anything at all. Do not think otherwise because it just does not happen. They do have automated systems that monitor aircraft arrivals and departures but not land and sea. None of them are particularly coordinated across the countries.

Also, as you probably know there is no internal passport control within the Schengen area. if you really want an exit stamp you will need to ask, and they do not always even have them in some places.

Thank goodness it is that way. My passports now have a chance of lasting until renewal time, which they never did before Schengen.