Firearm transfers initiated by non-FFL sellers are the ones being targeted by the various legislative proposals. They don't want people to be able to complete private transactions ANYWHERE they may occur - gun show parking lots, backyards, your neighbor's kitchen, the duck blind, etc. They want all transfers approved, recorded, and tracked by the government.
The arguments begin when you start trying to workout the details - the who, what, where, when, and how - of actually accomplishing this 'Universal Background Check' program. And, of course, there are many who feel that government involvement in any aspect of a firearm transfer is unconstitutional.