There are actually plenty of good (and legitimate) reasons to buy property in such a way the keeps the owner relatively private.

1. Keep your asset from appearing in a quick asset search. 2. Keep John Q. Public from finding your address via Spokeo, or similar data aggregators. 3. ..?

Same applies to a lease, be it commercial or residential. Most of the cheap background checks used by property managers are cheap because the company (say Experian, etc) receive your information for their product in return. The way around that, is to rent the property through a shell, or to a trust.