Well, I see lots of people making more or less the same argument to you TG, and your response being 'nuh-uh'. Sorry, but that's how it sounds to me. So, what the heck, I'll take a crack at it.
This has nothing to do with gun shows. This is a private party transaction. It can happen at a gun show, at your house, or anywhere else a buyer and seller meet. Heck, a couple weeks ago I bought a 1911 off another member here. We met in a parking lot. It occurs frequently at gun shows because there's usually lots of gun folks there.
Now, what is required of the participants in this transaction, wherever it happens? Varies state to state. I'll tackle Michigan because that's where I am. If it's a rifle or shotgun, I give the guy my money or beaver pelts or whatever else he wants to trade for and he gives me the gun. End. There is no paperwork before or after the sale because Michigan doesn't require it. You don't even have to know each other's names, let alone do a check.
Pistols are different. Private party sales are perfectly legal, but before buying Michigan requires that the purchaser has either a purchase permit (good for 10 days), or a CHL. The police do a background check when you get either of those. The sale also has to be recorded on transfer form. Seller and buyer each get a copy and one goes to the State Police to remove the gun from the seller's name. (Michigan does require registration of pistols). Purchaser then has to take the pistol and the transfer form in to get it registered. Of course it does behoove both buyer and seller to check each others' IDs before putting the names on the form.
So, in the case of pistols, in Michigan, there is absolutely no difference in the checks that are done whether it is sold by an FFL or a private party.
Rifles and shotguns, of course, require a NICS and a 4473 when purchased from a dealer. So that's the extent of the 'loophole'. People can buy a rifle or shotgun without paperwork. It is, of course, quite illegal for them to do so if they are prohibited by law. And prohibited persons that have fake ids or straw purchasers can also get them. Also illegally.