DNA matches for those of us who are trying to use it for genealogy are like gold. There is nothing more frustrating than when you have that match and the person never answers. There is not a lot you can do about their lack of response, but if you know how, you may be able to find out how they match you.
First of all, the match has to have a name, and they have to not have a common name like John Smith for this technique to work. Second, I really feel to be successful, you must have a fairly robust tree, and by that I mean you need to try and fill in the descendants of the siblings of your ancestors. For the best results with DNA matches, you really need to get back as far as you can, at least to the 5th and 6th grandparents, but larger matches will be closer, so you don't have to go all that far. Lastly, it is really helpful if you have someone that matches in common (shared matches on Ancestry) that you have identified, because then you know exactly what side (maternal or paternal) you need to look for. That's not completely necessary but it is helpful.
Oh, and you probably want to be confident in your internet skills. I have been using the internet to find relatives and their phone numbers for over a decade. Back then, if I had a question I would search and then call, now I reach out to them on Facebook if I find them there. Oddly enough, I have had pretty good success with reaching DNA matches that way too. Oh, and expect it to take some time too. It generally takes me an hour to several hours to find a match.
So say your match Vernon Pumpernickel is matching you at a 3rd cousin level (49 cM). First I would look at the shared matches, and I find that Vernon is matching two people from Granny Smith's side of the family. I search my tree and alas, no Pumpernickel's are in my tree. So then I go to google, and I search "Vernon Pumpernickel". I especially love it if Vernon's listed in an online obituary, but we aren't always that lucky. My next choice is if he's listed on a genealogy message board, but again, that doesn't always happen.
Vernon does show up on those sites that give "background checks" but you don't have to subscribe to them to get information. You click on the link and it lists relatives. Not how they are related but relatives. You find the names Norma Pumpernickel and Barbara Pumpernickel as relatives of the only two Vernon's listed. So try a search for them on Google and on ancestry for the places that Vernon lives.
Ancestry gives you a social security claim index on Norma Pumpernickel with names of John Moses and Grace Adams as parents. You find an obituary for Barbara Pumpernickel and it says she was preceded in death by her parents Norman Rye and Helen Smith. Back to my tree, I would search for the names, and if you have a robust enough tree, you just may find that Helen Smith is listed as the cousin of your Granny Smith, but you never checked further than that. So back to Helen Smith and searching for records for her. Bingo, you find out that Helen Smith married Norman Rye in 1908.
It's not always that easy, and the more in common matches you have, the more certain you can be that you really found the match. The next thing I usually do is send them a message saying I think I found our connection. Oddly enough, you are more likely to get a response from that message, even if you did do a bit of internet stalking to figure it out.
Desperate times call for desperate measures though. And yes, I am a true genealogy addict, and if it's a large enough match, I am desperate enough to research a match just to figure out how it connects. Triangulation on Gedmatch is the ideal way to verify matches, but you know, it's hard to do when people don't want to put their kit on Gedmatch.