Democrats are going for full gun registration.
Any law that bans private sales of firearms will necessarily require gun registration. Let's assume that I own over 20 firearms. How will the police be able to tell which of them I bought with a background check and which I did not? If there is no list of 'pre-background check' firearms, I can simply claim I bought the guns prior to the ban.
Possession would also have to be deemed the equivalent of ownership, as it is currently for the prohibitions against felons carrying a gun. If I have gun in my possession and the police demand to see evidence I passed a background check, I could simply claim it is not my gun, I am simply 'holding it for a friend'.
In order for such a law to work, every firearm in the country would need to be registered with the government. The government can then check the list of legal guns against what I have in my possession, and charge me with a crime for any discrepancies.
This is an insanely dangerous and likely un-constitutional law.
It would instantly turn tens of millions of law abiding gun owners into felons, as registration rates would fall well short of actual ownership.