You can see if trees are private before you look at the match. There is a little lock beside match names with private trees.
Sometimes these matches do have ancestry trees that are not linked to the dna test. You can look at these for clues.
My tree is Private and will be until it's documented to my satisfaction. There is so much misinformation on ancestry that I don't want to contribute to it. I also have a family line that is uncertain. I have shared my tree freely with anyone who asks. When I do, I can caution my matches about documentation and any other issues.
My experience with matches who have Private or No Trees has been very positive. I have emailed all of my Moderate or higher matches with private trees and many of my Low or Very Low matches. 90% have responded in a day or two and most have shared their trees as I do with them. I have also emailed people with Public trees when I find a common ancestor.
It only takes a minute to email a Private Tree owner. You may be surprised to find how accommodating he or she is. I have met the nicest people. Please don't give up on your Private Tree matches. You may be very pleasantly surprised.