I do keep my tree private for the general reasons you have mentioned, but if anyone contacts me, I respond each time. I have not been as successful when sending inquiries to private trees.
Part of the reason is that I have some things that I am sure of to some degree, but just as important when I have contacted people or helped, before being more careful, they have taken the information and just attached it (even when it was not a match) and then it eventually comes around in trees and I waste time trying to correct or research the mistake.
Had a couple of real bad instances. There are a couple of names in my tree that are very far from common which never showed up. I passed information on (in one case to an owner of a private tree that has 40,000 or whatever amount) to find out whether there was a connection. They told me know, different family, then a year or so later the name shows up in the tree.
It seems to happen when people seem to be "name collecting" rather then researching.