Wrong on both counts. The government owns the building, and citizens can indeed be prohibited from entry onto government lands. Just try wandering into the Naval Special Weapons Depot in Virginia sometime...
As a parallel example, I own Ford stock. I may not, however, legally wander into a Ford plant, or take a new Ford automobile from the line, or take parts, despite the fact that I OWN part of the company.
You can be prevented from entering via security or other requirements but if you entered you would not be trespassing. You'd be breaking and entering or violating some other rule. You cannot trespass on what you own. Go to the Depot and read the signs. I'll bet they say "No entry. Authorized personnel only" or some such. Not no tresspassing. (At least the ones in the bases around here read that way.
As a parallel example, I own Ford stock. I may not, however, legally wander into a Ford plant, or take a new Ford automobile from the line, or take parts, despite the fact that I OWN part of the company.
Close but no cigar. The government does not own the school, we do. We again, are not trespassing. Now if the officer took something other than pictures he would have been stealing from all of us. He didn't take anything and he didn't trespass. No problem.
(I know what you mean about Ford. Bummer isn't it. I own Green Bay Packers stock and I can't ven get tickets)