Originally Posted by
Alex71
Our kids have both American and German citizenship, which means that they have to enter the US on their American passport and at enter and leave the Schengen zone on their German passports.
That is indeed the case if wanting to keep things as simple as possible. But even CBP provides info about dual-citizens of the U.S. using an ESTA/VWP passport to travel to the U.S.; and it suggests that, on arrival, such passengers present themselves as U.S. citizens -- even if/when without a currently valid US passport. With a minor's expired US passport to present, it becomes much simpler on arrival in the U.S. even if having traveled with an ESTA/VWP passport, as is well possible.
The U.S. citizenship of young minors who are US citizens is not subject to lawful relinquishment/renunciation of US citizenship. It's not even for older minors and adults, merely on the basis of use of a VWP passport to travel to (or even enter) the U.S. This area has been extensively deliberated as part of what was called the GWOT, especially as various kinds of targeting were subject to such considerations of legality tied to citizenship status.
Originally Posted by Alex71
We once gave the check in agent in the US their German passport, but were asked for their American ones, since the German passports apparently created an an error message when pulled them through her passport reader.
My US-EU dual-citizen relatives do a large number of annual trips using both passports on one-way or return trips originating on either side of the Atlantic; and they have never had the kind of issue you mention above as long as the passport info was updated in the booking prior to an airline agent in the U.S. swiping the passports, non-US or US.
When the passport info in the booking doesn't match the passport info from the agent's swipe of the passport, then airline systems often alert. Even if using just the passports of one country and not being a dual-citizen, they can alert due to different passport info.
Originally Posted by Alex71
Apparently, one has to leave the US on the same passport one enters it on.
No, but it may make things more simple. We have a lot of people leaving the U.S. on different passports than they used to arrive into the country and it's generally not a problem for them when checking in and flying out of the country. If they came in on a VWP passport when using an ESTA and don't leave on the same passport, then the odds of a problem arise. But still not generally a show-stopper.