It varies with the area, but there are usually some non FFL sellers
An FFL (Federal Firearms License) is required for all persons "engaging in the business" of dealing in firearms. This is defined administratively by the BATF, but basically it means that someone who deals in guns with the idea of making a living at it needs to be a licensed dealer.
Individuals, even those with large gun collections may sell part, or all of their collection on an ocassional basis without being considered a dealer. I'm not sure of the actual number the Feds might use, but too many guns, and on a regular basis without a license can get you in trouble. There is some leeway, mostly determined by individual situations, and their interpretation of your intent.
You can go to every gun show for months, and sell guns from your collection at each one and not have anything to worry about, but if you are buying guns at a show and turning around and selling them (for a profit) you can run afoul of the law. People have been "entrapped" and charged at gun shows after buying a gun, by having an undercover agent offer them more than they paid for it, and when they sell to him, he busts them for "engaging in the business" without a license. It isn't right, and usually doesn't result in a conviction, it isn't common, but it has been done to people. Your tax dollars at work.
Private (non FFL) gun owners go to gun shows to buy, sell, and trade guns. That's why we have gun shows. FFL dealers at gun shows are under the same rules as they are at their shops. That means filling out the forms and doing the background checks. The so called "gun show loophole" that the anti gun people and pandering politicans want to "close" is nothing more than the legal sale of guns at shows by private parties (non FFL). People selling some of their collection, or one gun trying to sell his deer rifle because he is out of work, to the antis they are all "unlicensed gun dealers". Our historical right to do as we please with our private property is under assault, and if they get their way, the only people at gun shows will be FFL dealers, and real quick the shows will wither and fade. One local show near me will likely go away for good, due to the zealous efforts of the local deptuy DA, who (since it is held on public property) is requireing every seller at the show to hold $1 million in insurance for the durations of the show (3 days). Every seller. Not just the gun sellers, but the people selling jewelry, the people selling t-shirts, even the hot dog guy! The insurance costs a couple of hundred, and many small vendors don't even clear that much profit even when they have a "good" show! So, between table rent, and insurance costs, they are unlikely to make even a handful of dollars for all their time and effort. Loading up everything, spending 6-8 hours Sat & Sun sitting at a table, then take whats' left back, and earn maybe $20 clear? Most folks won't bother anymore. Sad.
the form you fill out when buying through a store is not a registration and is not kept by the government
Yes, and no. Form 4473s are kept by the dealers, so, yes they are not kept by the government. But they are open to inspection by the BATFE during an investigation, AND, when the dealer goes out of business, ALL the records must be sent to the BATFE, and are kept there, forever.