This "make background checks mandatory for private sales" thing sounds good, but won't work. It won't work for the same reason that no one pays sales tax at a garage sale: you're supposed to do so, but there's no way for the government to enforce the sales tax laws on people who don't hold a business license.
The existing background check system works because it's tied to firearm dealers' licenses: they've got to do it to keep their business license.
Ironically, during the Clinton administration the feds went on a "too many people have FFLs, let's make them much more expensive and hard to get!" spree. Which now means that many fewer people participate in the background check system, as a result of another initiative that sounded good to people who have a tenuous connection to reality.
For what it's worth, if you do go buy a firearm on the internet, odds are really good that you're getting a background check anyway. Why? Because to ship a firearm, it's got to go from FFL to FFL. And the FFL in your town handling your shipment is required to do a background check.
But, it sure does sounds good to propose such a law: to people who have no clue how things actually work. Which, it turns out, is true of most of the "feel good!" solutions non-gun owners concoct to impose on gun owners. Comes of trying to legislate to match what they see in movies and in cop shows rather than what actually happens in reality. So, I wonder how this study came up with their numbers. Did they just say "hmm, X% of people buying their guns person to person commit a crime, a BG check would magically change that number to 0%"? I suppose it might, if 100% of the people followed the new, easily ignorable law. Considering that they're going and ignoring other, stricter laws to commit their crimes (like, "killing people is illegal"), that sounds rather optimistic.