Our local shows require private individuals with tables at their shows to follow all State firearms laws (waiting periods). That means at the show, the seller and buyer exchange information and FOID numbers (hey, its Illinois), and usually some sort of deposit is held by the seller. There isn't any requirement for a background check or completing the national registration form (yellow BATF) between private sellers. The seller can technically only transfer the firearm to the buyer after the waiting period (24 hrs. for a long gun, 72 hrs for a handgun). At that point the buyer and seller meet somewhere mutually agreed upon and make the final exchange...balance due gets paid, and buyer takes firearm possession.
Some shows here even provide a two part form for this exchange of information between buyer and seller. It's becoming rare to see gun show attendees (without a table) here in Illinois making private to private transactions. In fact, the gun show promoters are getting very nasty about that.
Most gun shows that I've been to outside of Illinois are much friendlier with private sales, whether they're table sales or attendee sales. Most are cash and carry transactions between law abiding show attendees, without any paperwork required.
Personally, I've had enough of the red tape at our local gun shows. Promoters are beginning to make gunowners feel intimidated by all of the attention they've gotten. Making announcements over the intercoms, passing out restrictions which aren't even applicable to State or Federal law. In addition, prices here have climbed beyond reasonable. Shows are over crowded, and people seem tense. Conversations between people in parking areas are quickly dispersed by gun show police, as if some major arms transactions are going down.
It's all becoming a big joke as more people take interest in buying a gun make their way to these shows, and find themselves strangled by the promoters self-protectionism.