In my neighborhood, there's a public and a Catholic high school track, both used by the public, with no one objecting.
In fact, some years ago when they rebuilt the public school track with a rubberized surface, they openly allowed the public access to the track--i.e. used to be they kept gates closed; now they have them open and don't prevent people from running/walking there unless there's a track meet or football game. (In fact, I seem to remember that they had some kind of welcome event to let people know the track was available to the public--or in some way announced its opening--and they have rules for use and a sign saying you use the track at your own risk, etc.) I've even run there during school hours. If the track is being used for a PE class or track practice, I'll approach the teacher or coach and ask them if they mind if I run there. If they say no, I leave. But they almost always say yes. I know the coach and one or two of the teachers, so that might help--plus, I'm careful to avoid the most traveled lanes.
The Catholic school track is paved, less appealing than the one at the public school, so it's generally my second choice. But it's okay for short runs and closer to home. I definitely won't run on that one during school hours, but will use it some evenings or weekends. There too, many from outside the school use the track with no one objecting. I don't think the school made any announcements about public use either way, but they never chase people off.
In another area where I lived, one Catholic school I called about using the track was a lot less receptive. The person who took the call acted very huffy and said it was private property--but she transferred me to the athletic department where the guy who answered kind of sneered at my being a woman and wanting to train, very condescending. So I decided not to pursue the matter. Yet at a nearby public school, the woman who took my call was quite friendly--"Oh yes, it's fine if you want to run there, if there's not a game--let me check and I'll call you back." And she did call telling me to come on over. I get that the Catholic school in this case was within its rights. But the difference in the way I was treated when I called the two schools really struck me--if this is the vibe that particular Catholic school still puts out, then their staff definitely needs better communication skills. How would they know I wasn't planning to send my kids to that school--and might have decided not to based on the response I got?