Building permits and trademarks are required to be public. Other records are not required.
Consider corporate ownership. Some US states allows anonymous corporate ownership. (See http://blogs.reuters.com/great-debate/2012/10/09/its-time-to... complaining about such companies.) If it's okay to own a company without that information being public, then why not a domain?
Such companies are used as shell companies. If you want to trademark a name but not have others know that it's you, then start a Nevada LLC, owned by a Delaware LLC, owned by a company registered in the Cayman Islands, owned by you, and have the Nevada company take out the trademark.
Also, the term you want is 'hidden from the public' not 'hidden from the public domain.' "Public domain" applies to, for example, works that were under copyright but are no longer so.