This depends greatly on the laws of your state and how the campus has contracted out, etc.
I went to Iowa State, and back then they had campus security. They were essentially "rent-a-cops" - security guards. The campus was state property and therefore not part of the city and therefore the city cops had no jurisdiction there. Having said that, the campus security could "deputize" them, if needed and authorize them to come on campus. This was, however, a major sore spot for the city police. A few years after I graduated, the state board of regents decided to "outsource" campus security. They took bids. The city police force put in a bid, and won. So now-a-days, "campus security" and Ames Police Department are one and the same.
Oddly enough, the married student housing was organized as an unincoroprated town. They had a "Sherrif's Department" of their own (separate from the county Sherrif). Never understood quite how that worked. They were considered law enforcement, like any other police or sherrif's department, whereas campus security was in the realm of a security firm.
If you observe a crime in your jurisdiction, you can follow the criminal beyond the borders of your jurisdiction if you are in pursuit. Another jurisdiction can temporarilty deputize you - effectively making you policemen in their jurisdiction.
If a crime occured on campus, the campus police could elect to call the local police for backup, thus deputizing them. But typically they would not (unless they are out numbered, out gunned, etc. - I'm sure in a S.W.A.T. type situation, the campus police would call someone in). Your campus may have a standing agreement with local police, deputizing them on campus property.
Here's where the splitting hair of law enforcement (police, sherriff, etc.) vs. security lies. If you are a policeman, you are authorized to act as a policeman at all times. If you are off duty, you are still able to act as a policeman. Many a police department requires their off-duty officers to carry a gun. If you are security, you are not authorized to act in a police capacity when off-duty. You are only authorized to perform your police type functions in your jurisdiction (barring pursuit), while you are on-duty. You would have authority over anybody within your jurisdiction - student, non-student, faculty, etc.
Your campus is most likely state property, but, of course, the state police have jurisdiction there too.
What are your state laws like? Is your state owned campus considered state property and not part of the city (probably, since you have campus cops)? Does your campus have a standing deputizatoin policy with the local police?
You know what. If you obey the law, none of this matters anyway.