I know that the Muskingum University police were a bona-fide police force, with authority to arrest, issue citations, and all of the other police-like stuff.
Of course, they were a complicated matter because despite being a legitimate police force, they still had to answer to University administration. At the risk of saying something political, that was always a caveat that greatly restricted their usefulness. Of course, New Concord police still retained full authority to make arrests, issue citations, and the like on the Muskingum campus, so things had a way of working out.
Kings Island police are likely in a similar situation, having the legal powers of a police force, but still answering to the authority of KI/Cedar Fair administration. And in much the same way, there is still Mason Police, who would be unrestricted by park administration.
Everything in this post is speculation on my part, lest anything come off as my own knowledge. Basically, just providing an analogy to which I am familiar.