Yes they would. My college went from private security to using police. A time honored prank was to sneak into the museum in administration building at night and move things around. If the students got caught and had done no damage it was treated like any other prank. But the first year of using police a couple of students were charged with felony burglary (because of the value of the clothing the mannequin they moved was wearing). The school did NOT want to press charges, but because the security officer who caught them was an actual police officer they had no say in the matter.
So yes, school cops do have to report things to local police because they ARE local police.
Ignorance fought. It looks like officers employed through them do work directly for the district, though I note that the org chart for the KCKPD itself does have SROs under their Community Support Division. I don’t feel like digging enough to see what the exact relationship between KCKPD and the KCKPSPD (oof) is like.