It depends on the person that is selling it. You will be able to easily tell if the person is just a collector of guns trying to sell off some of his lot or an actual firearms dealer. There will usually be a banner that shows the corporate affiliation of the person.
As for a private party sale, unless you're buying from a firearms dealer or a company, yes the gun will be a private party sale. All persons who sell guns at the gun shows I have been to have to have a FFL license in order to display their lot and make sales. If you are out of state, you will have to have the gun shipped to a registered FFL dealer in your home state.
On your part, unless you're buying for a company or dealer, it will be a private purchase. Depending on which state you reside will decide how anonymous the sale truly is. Many states have a data base for persons who have purchased/transferred a gun through a FFL dealer, which I believe is an invasion of privacy and an abrupt disregard of the 2nd Amendment.