This little video is from a search I performed on Ancestry.com today.
The information in the public tree is private and does not show. However, the "private" information does show in the search results for this person.
So is it possible that something in a public tree which is marked "private" may appear in the search results?
It did in this case.
[update--apparently when users create public trees they can set it to show the information in search results, but to show "private" when viewing the actual items.]