Maybe schools could rent out their empty rooms to the private schools and share common resources. Kind of like "educational infrastructure as a service". I'm only half joking, I've seen some real examples of this: elementary school/alternative school, high school/deaf-only school, high school/adult school. The buildings were divvied up, and the admin and everything else was a different organization. The only complicated thing was having to have separate entrances for each and locked doors inside so students can't pass from one side to another.