I'm looking for my dad's birth family, so I understand your desire to connect with folks! I haven't kept score, but I'm guessing I've been fairly successful at getting responses, and I've been invited to a few trees. One of the reasons I hear about keeping trees private is that they are not confident in the accuracy of their trees.

One of the things you could do is Google the match's user name and see if there is evidence of similar email addresses on the Internet. I've found quite a few direct email addresses this way. If they have posting histories with genealogical sites, even better.

If it is within your budget, you should also get tested at FTDNA and 23&Me, so your pool of matches widen. I've had very good luck connecting with FTDNA matches (an email address is provided on lots of profiles), and my 23&Me test is pending; however, it is my understanding that the 23&Me folks are less likely to respond to inquiries because 1) you have to reach out to a messaging system that lots of folks have not set up to notify them when messages are received; and 2) it is more geek oriented than genealogy oriented.

You also might have better luck uploading your raw results to GedMatch, where most folks (from FTDNA, Ancestry and 23&Me) make an email address available, and seem to almost always respond.

Good luck!