This can be a big pain, but there are a couple of things I do when I see a potentially useful private match. One is to click on their Ancestry id. Sometimes they have both private and public trees. They may also have a message board posting history that can provide leads. For any private match that is moderate confidence or above, I send an email giving them information about my family and requesting access. I get very few responses, but sometimes it pays off. Finally, I google their Ancestry id. Sometimes I find that they have trees or postings on rootsweb and other public sites.