Earlier quoted context omitted.
Wait till you hear about privately-owned company and railroad police departments...
Do these private police have more authority than the private security hired by other businesses?
Private universities, same as public, may have Campus Police instead of Company Police, and those officers also have jurisdiction on public ways passing through the campus. In some cases they also have one mile extraterritorial jurisdiction; moreover, they can receive broader jurisdiction through agreements with the city/county law enforcement.
Railroad police are to be certified in their home state and have nationwide jurisdiction by federal law, on the property and rights-of-way of their employer as well as in connection with its services (conceivably quite a wide scope as railroads run right through virtually all major cities--the CPD "bait truck" incident that cause controversy a few years ago involved NSRR Police). With the exception of Amtrak, these are all private companies.
I'm not sure to what extend qualified immunity does or does not apply to them, however.
PA definitely has something like this too.. just like in NC, it originated from company towns, in their case mining.