The paperwork done when buying a firearm from an FFL is not a registration. The government is not provided the serial number as a part of a sale, in fact they d

The paperwork done when buying a firearm from an FFL is not a registration. The government is not provided the serial number as a part of a sale, in fact they don't even find out wether or not you purchased the firearm. Similarly, if you get a rejection during the NICS background check the dealer doesn't find out why, just that you are not eligible to purchase. Both parties are kept blind to information that they shouldn't be privy to. Unless you live in a state that requires registration of firearms, the government doesn't just find out who owns one. That isn't to say there isn't some legal mechanism by which the government can get that info from the shop that sold it. If you truly want an anonymous firearm, buying one in a face-to-face transaction is the closest your typical person will reasonably get, and that still requires you to show some form of ID (a drivers license or a carry permit, typically) to the seller. At least that's how it works in South Carolina.