Look at the Bigger Picture.
One very big factor involving private purchases that people don't often discuss or even bring up, is how the buyer finds the seller and what are the circumstances around the actual transfer. The seller, if he/she wants to help prevent their gun from getting into the hands of a prohibited person, has many avenues available to figure out if the buyer is legit or not. Of course, I agree that none of them are 100% fool proof. However, a little caution and awareness can go along way.
For instance, if I post a "gun for sale" card at my local gun club and receive a response, I can immediately ask, "Are you a member at _ _ _ _ gun club?" If yes, you can probably feel a little easier about the sale right away. At the very least you could check with the club leadership to see if he's telling the truth. If he says "no", then ask how he became aware of the gun for sale. A little honest questioning should be no problem for someone who is legit and is interested in the gun for lawful purposes. If you get someone who gets uptight or uneasy, then your radar should be on.
If you post an add in a newspaper and receive a response, one would have to be aware of how the potential buyer wants the transfer to occur and where. A few simple questions wouldn't hurt while you have him on the phone. "Are you legally able to purchase a firearm? Are you 18 or older? How will you pay for this? Are you willing to provide name, address, phone number, and a valid ID? I require you to sign a bill of sale with your name, address and phone number. Are you OK with that?"
These are just a few of the methods to try and gain at least some insight into a prospective buyer. Again, nothing is fool proof, but you can at least reduce the risk. If you are a legitimate buyer, why would you object to a few straightforward and honest questions? I'd be prepared for such if you are conducting a private purchase and you are the buyer.