If they don't have a permit, that's what the bill of sale is for .
That's a horrible idea. You are not required to have a permit. A Bill of Sale is a pre-registration document. What keeps the seller of providing your info without a warrant? Once the LEOs have your info on file, your weapon is registered.
What's to say that someone may sell you a gun, demand this "Bill of Sale" and then burglarize your house as he now knows you have guns and where you live? When you sell a car and provide a Bill of Sale, which party gets the Bill of Sale?