I don't know if it's legal or not, but I do think that selling and shipping a firearm to someone that you don't know is like walking way out on a limb and placing your head on the chopping block there.
When you involve an FFL holder in the transfer, the transfer goes into his records and it becomes his responsibility to ensure that the purchaser is legit and is legally entitled to own a firearm.
Example: Felon Joe Diddly buys your gun via auction. You receive payment and send him the firearm. Upon receipt he loads up and heads for the local post office to settle a dispute with the post master about his Prozac/Lithium prescription not coming in on time. Lots of dead people result. What happens? Who's at fault? Even if the DA takes pity and doesn't charge you with transferring a firearm to felon, you have the victims family lining up to sue you for..... I forget what it's called in civil court... same thing that OJ was sued for after the DA fumbled the ball in that case.
Bad Juju in general.
Ian