How does the local school district know if you have school age kids and whether or not you've registered them?

It largely depends on the State. In know in my state of Virginia, and those like it, you don't have to report your children to the local school district unless you want them to go to public schools.

They really have no other way of knowing.

Some may say that the census could be a way, but you also have to take into account that people move all the time. In fact, I was in the middle of moving during this most recent census.

Some states have a requirement by law, which says that you're obligated to register your school age children with the local school district.

Census records