It's a big can of worms, and fly's in the face of reason. The people who register, are part owners of "the papers."
The right of the people to be secure in their persons, houses, papers, and effects, against unreasonable searches and seizures, shall not be violated, and no Warrants shall issue, but upon probable cause, supported by Oath or affirmation, and particularly describing the place to be searched, and the persons or things to be seized.
Yes, mine is a weak argument, but the whole idea of giving such sensitive information to others is unreasonable, without probable cause.
Consider the ways criminals may use such information once made public.
Why do you (the public) have the right to know what I own? Do you also want to know the color of my underwear? Boxers or briefs?