Have you ever been to a gun show? The people who rent booths at gun shows are gun store owners. Anyone selling guns that holds an FFL (federal firearms license) has to run a background check for every sell, even at gun shows.
Now if a non-FFL holder wanted to rent a booth and sell their personal collection of firearms they could do so without running background checks. But if this person bought/sold guns regularly, or bought brand new guns and sold brand new guns, they would be violating the law. What they are doing would be considered a straw-man purchase and is illegal.
However, I suppose a mentally deranged person, or potential domestic terrorist, could hang out at the gun show and solicit their fellow visitors to sell them some of their privately owned guns. That could happen, legally, without a background check. I invite you to go to the next gun show in your area and see if you can get any action on that play.