There is no such requirement for long guns.
I'm pretty sure you can run your own background checks when you sell a firearm, through the same system a licensed dealer does, via an ATF form 4473.
If no permit, then the seller may want to get have a background check performed on their own. Usually about 40 bucks from any number of places. This does not release you from liability however.
You could ask the local PD, but they can be reticent to accept this responsibility.
You can ALWAYS go to a licensed FFL and do the transaction there. There will be fees (some stupidly high, shop around), but doing so will release you from liability.
Selling a firearm to a private person is dangerous. Namely in that you, the seller, have all the burden of insuring that the purchaser is legal (In your state). If you sell a gun to someone who cannot legally posses one, you have committed a felony. Period.
Virtually every state discourages it, and some go out of their way to make it seem illegal.
Forgot about the ATF form...
Like all laws, gun laws can be murky. For example, in my state I think what it actually says is that the buyer must have a permit or CCW for a person-to-person handgun purchase. It doesn't explicitly lay out the seller's responsibility to verify that the buyer has one of those, though. It also doesn't specify the seller's responsibility to document any of that or maintain any type of record of the transaction.
There's no shortage of stupidity in the anti-gun community. They're going to look for "problems" with gun laws in every area they can possibly find because in the end, they aren't just trying to regulate sales, they're trying to tip toe towards a full scale gun ban. That's why we can't give ground to the Brady Campaign no matter how much they whine that we aren't willing to "compromise", because too many of our gun rights have already been "compromised" in the past and if we keep "compromising" then we can expect to have nothing left when all is said and done.
I like how the article cites "the victims". What victims?? When was the last time a .50 BMG was used for a crime?
This is like setting up an illegal drug deal online, then complaining that it was legal because it happened.