To emphasize a need for the search box to be isolated on the Visitor page, the visitors search box on my member's only site still allows a person to see stuff. For example, by entering admin as a search term, a person can see links to any member who has used that term in their posts. If the visitor clicks on my name, the sites says restricted entry, but!
Every module (or block?) that hasn't been tagged Members only will appear in the sub menu and all, or part of that that module is freely accessible.
As Cheetah has hundreds of places to mark things Member's Only, mistakes will happen.
The sure-fire way of stopping this is to eliminate the search box from the visitor home page.