Police officers within their jurisdiction do not need a permit to carry a gun. The method or conditions of carry are set forth in the department regulations.
Usually, officers are required to carry their duty firearm or a compatible (in ammo) firearm at all times, on or off duty. Again, that is in the department regulations, but the point is that a sworn officer is never really off duty; he is required to respond to a crime in progress no matter where he is.
Police officers are not bound by restrictions on carry that have been established for individual concealed carry license holders.
Carry outside the jurisdiction is trickier, and depends on state law. Most states will allow an officer to carry anywhere within the state, but some have restrictions. Outside your state, it gets iffy. Most police will honor a badge and give the bearer a pass on carry, but some states will not do so for off duty police.
The worst case may be New York City, where NO outside police officer (even NY state police) may carry a gun, on or off duty, for any reason. Even police picking up prisoners for transport to other jurisdictions must be met by an NYC officer, who will take custody of the "foreign" officer's gun, to be returned only when he leaves with the prisoner.
Just sort of FWIW, I detect that you have not gone far in your effort to have a police career and you seem to be going in that direction only so you can carry a gun. If so, forget it. Police work is hard, and discipline is rigid, particularly for a rookie. The picture of the movie cop, with thrilling car chases and hourly gun battles, is nonsense. And it will be a long time before you do the CSI bit. That is not you bending over the body; that is you directing traffic and trying to not look too much like the newbie you are.
That is not you asking probing questions and arresting murderers; that is you wrestling drunks and cleaning up the vomit in the back of the patrol car. Taking a ration from the "honest citizens" is bad when you can't do anything about it. I might slug some SOB who calls me names; you can't. And you also have to take a ration from your sergeant, and everyone who is above you, which, while you are a rookie, is everyone in the department.
Be prepared for tough physical training, lots of boring classes in law, police procedures, interaction with the community, race relations, first aid (yes, delivering a baby is included), and sometimes a foreign language. Shooting is a very minor part of your training and of the knowledge you have to have to be a good cop. Lots of luck.
Jim