Most private sales are between friends and familly. The only way this law could have any teeth is if all guns are registered first.
While I certainly do not condone required NICS for all transfers, the idea that registration is needed to make people comply with the law is flawed. It is unlikely that after all this many people would be ignorant of the fact that such a law has passed. A transfer then requires two people willing to break the law or ignorant of the law to occur. This flies in the face of much of the we are law abiding citizens arguments that are made, if all these law abiding citizens are now breaking the law.
It would work because a majority of people are law abiding. This is not to say there wouldn't be people who, out of ignorance or misguided principle, would not obey the law, but they would be in the minority.
An interesting consideration; If some sort of universal background check legislation were to pass, would it address and revise the current exemption for certain states' ccw permit(or similar such as the case of NE's purchase permit), create a legislative conflict(such as the legislation explicitly says all transfers, but doesn't create changes in the previous Brady language), or just leave it be. If left with the exemption, for permit holders...then presumably private sales could continue "unchecked" between permit holders as they are exempted.