18 USC 922(q)(2)
(A)It shall be unlawful for any individual knowingly to possess a firearm that has moved in or that otherwise affects interstate or foreign commerce at a place that the individual knows, or has reasonable cause to believe, is a school zone.
(B)Subparagraph (A) does not apply to the possession of a firearm—
(i)on private property not part of school grounds;
(ii)if the individual possessing the firearm is licensed to do so by the State in which the school zone is located or a political subdivision of the State, and the law of the State or political subdivision requires that, before an individual obtains such a license, the law enforcement authorities of the State or political subdivision verify that the individual is qualified under law to receive the license;
If the state allows carry in school zones and in the school itself, and the state does not allow school districts to prohibit carry, then teachers, aides, janitors, parents, etc. can legally carry in schools.
In Utah it is relatively common for people, including school staff, with a permit to carry in public schools. If one has a Utah CFP (concealed carry permit) it also allows the carrier to open carry in public schools. Public schools have no authority to prohibit firearms. Private schools can, of course, set their own policy but it is not illegal to not abide by the policy. Failure by private school staff to abide by the school policy, however, may result in disciplinary action up to and including termination.