That varies from city to city. Some have civil pilots and police officers as copilot.
In New York and LA you have to be a police officer first, before they train you to become a pilot (I think Philadelphia too).
Nevertheless, as long as there is no TFR in effect, they have the same rights and obligations as you and can only ask you to do things (stay away from a pursuit or crime scenes etc.). All that of course applies as long as you don't break the law, but even then, "intercepting" another aircraft by regular police is questionable in my opinion.