Law enforcement doesn't need a record of who owns the guns.
Nor does anybody else.
I have to register my car in each new state I live in, keep my insurance up to date, have required inspections and submit to a driver's test to get my license. Driving a two thousand pound mass of metal and glass at high speeds should rightly be a big responsibility that everyone should take seriously, that should be regulated to ensure appropriate safety for everyone on the road. Owning a gun should require a similar level of responsibility.
Anyone who wants to own a gun should likewise take a written test on gun safety, proper means of carrying, loading and unloading, and turning the safety on and off to get a gun owner's permit, though actually shooting it must be done with a licensed gun owner. After an appropriate waiting period, gun owners should take an actual test involving everything from loading and unloading, proper storage, even shooting proficiency. And each gun should be registered in each state it travels to, each gun owner should submit to an annual inspection for their weapon, and each gun purchase should come with mandatory liability insurance. The law should require that guns be stowed safely and require reporting of loss of theft. Gun ownership is a right, but with rights come responsibilities.