For privacy - Does one need both private domain AND private host?

[miraclesinc]

If one wants to keep personal details (address etc) private - is it enough to register the domain "private" - or can details somehow be gained through ones (non-private) hosting-company?

The reason I ask is that I see some services offer "Private domain AND hosting". I always assumed private domain was enough for privacy?

[miraclesinc]

I think I was a bit vague when stating my question.

What I mean to ask is this: Is there some way for the average (or slightly skilled) websurfer to find out the owner-details for a site which does have private domain, but not "private hosting"?

The reason I ask is that I see some sites offering "both private domain AND private hosting".
I always thought private domain was enough to keep personal details secure, but was I wrong?