I have been contacted by 2 people who had private trees and who saw a possible connection in my public tree - we have shared info and both are positive matches, although nothing new for either of us.
I have contacted 2 others myself, one has a private tree which they sent me an invite to and I found we are connected 2 ways.
The second person I contacted also answered - we both have public trees, and although we are 4th cousin/96% matches, neither of us can see a connection. We agreed to keep our match starred and contact again in the future if anything new pops up.
I am administering 2 DNA samples and trees currently, and will shortly add a third from another relative. I have about 1800 matches from my DNA, my husband has over 3000, and I expect the new test to be similar. It is a lot to keep on top of. I feel that I am an experienced researcher and detective, but some of my matches may not be- and may not have the time to devote to investigating the connections.
So far, I am not concerning myself with the private tree connections unless they are in the 4th cousin category (or better, if I ever get any). I figure my trees are public, and if the private matches are serious, they will look at my trees and contact me about a match. Maybe that is optimistic.