I think GJSchulze pretty much covered it, except that I believe that there is a database of that information kept in Albany. I know that our county office sends a photocopy of the purchase receipt to Albany, and I would imagine that all the info they key in (make, model, caliber, serial number) isn't just for printing out your gun card, but goes into a database too.
It can get pretty confusing, especially when registering milsurps. Who's the manufacturer of a Bulgarian Makarov? (We finally settled on listing the importer as the "make" and "Makarov" as the model.) What do you put as the caliber when you have a 9mm with a .22 conversion kit? (I was told to list the caliber I was most likely going to carry it with: 9mm.) What do you put as the caliber of a Nagant Revolver when there isn't room in the caliber column of the gun card for "7.62x38R"?
As far as I know from talking to the LEOs that are members of our club, they can carry department-approved guns "on the badge," meaning they don't need a state permit. I don't know how purchasing is handled -- I imagine that would vary by department. However, if they want to purchase and carry other firearms off-duty, they need to get a permit like everybody else. I believe this also simplifies things when they retire.