No.
We have that law in California. Of course everyone follows it
, just like speed limits and the laws prohibiting smoking in the park, leaving a child in a child car seat while going to the ATM 10 feet away, and having your dog safely restrained on the porch for a short time (it's also illegal to have the dog running loose).
It's just another tax on guns, no more, no less. That's true about all background checks. Hell, since I bought a slough of milsurps when they flooded the market last year, I already have more guns than I will ever bother to use. The state has no compelling need to check me out and charge me a fee to get one more gun -- they just want the money.
First off, there should be no checks. Guns should be sold like power saws, cars, lawnmowers, chainsaws, butcher knives, axes, baseball bats, lighter fluid, and every other dangerous thing you can buy at your local store.
That said, it's only the law-abiding citizen who will follow the law anyway. I'm sure that criminals DO buy firearms private-party. The black market IS a "private party" market.
The only thing this could accomplish is to add to the incentive to steal firearms. That endangers all of us.
A dangerous career criminal should be in jail. That's a solution that works, even here in California. It makes liberals and libertarians cringe, but that's where I depart from libertarian orthodoxy. The Three Strikes law works here, plain and simple.