personal sales or transfers
There is no Federal law that precludes the sale or transfer of a firearm from one private individual to another. The only time when the consideration of any kind of background check or paperwork is involved is when a seller has an FFL and is in the "business" of regularly selling firearms.
However, state laws vary in extreme when it comes to sales, transfer and posession. Witness California, New York and Illinois versus states like Texas, Georgia and Alaska.
I believe that the NRA or the NRA-ILA has a synopsis of all the states gun laws on their websites. Check it out.