There is no private sales loophole for gun purchases
Federal law doesn’t require private individuals to do background checks for gun sales
The federal law that requires background checks requires only licensed dealers to perform background checks -- not private individuals. A PolitiFact.com article described the rules:
Buying a gun has required a background check since the Brady Act took effect in 1994. (James Brady was badly wounded during the assassination attempt on President Reagan; he was Reagan's press secretary.) Under the law, federally licensed dealers must verify that a buyer has not been convicted of a serious crime or declared mentally incompetent or is blocked for any of about 10 reasons. Typically this is done online and takes less than a day.
But only licensed dealers must do this. The law doesn't apply to private sellers at gun shows, flea markets, or people who post firearms for sale on the Internet. If a private seller suspects that a buyer would be disqualified under federal rules, then they can't go through with the sale. But there is no background check, and no one needs to file any paperwork.