I can't remember what Ancestry call them but there are two types of private. One where people can just see the basic details on someone who matches. I use this on most of my trees and prefer they contact me if they want more information rather than just download all my tree.

There is another type where the tree is completely private and does not appear in searches. I use this in some trees, particularly one that I use for odd lookups, searches etc. If someone contacts me for further information in a couple of years time I will probably not even remember who the person in my tree is because it was just bit of random research.