In Texas we can trade guns as we see fit. I don't know how it works in other states. I guess you'd probably have to show identification if you bought one in a store, but I can't swear to it as I've never done that.
The instant check law (I forget the correct name for the act) requires each FFL to run a criminal background check for each buyer. The exact nature of the check is left to the state. Some states use the FBI, others have their own systems. They can charge for it, so it isn't an unfunded mandate.
The 1968 Gun Control Act made it a federal crime to sell any firearm to a person who resides in another state. This means there are no private sales of firearms across state lines, and the feds thus put themselves out of the firearm private sale business due to the commerce clause in the constitution.
Some states have a firearms purchaser card, required to buy any gun. New Jersey, Illinois, and Hawaii are examples. In these states, the buyer and seller have to complete some paperwork and possibly register the sale with the state. Most states do not require this, so if you want to buy your friend's gun in Texas, you give him the money.