Federal law- an unlicensed person may sell a handgun to a resident of their own state, if seller does not have reason to believe buyer cannot legally possess a handgun.
State law- ALL over the place. Some require the sale to take place AT the police dept. Some require a purchase permit (even for private sales)- and some require nothing.
Common misconception- that a gun is somehow "registered" to a particular owner (Thank you, CSI for that misinformation) In most of the US, there is NO registration of ordinary firearms. It is not "in your name" unless your name is Colt, Wesson, or Highpoint.
For a straight answer, suggest you check with local gun shop, who will probably have a better grip on state gun laws that local PD. As said, shops also do consignment sales, where they get an amount for selling for you.
When you buy a gun from a dealer, it is not registered to you in some mysterious manner (no one to register it WITH). The dealer maintains a record of the sale at his shop for 20 years. Those records CAN be subpoenaed by law enforcement.
What we have here is... failure- to communicate.