From my Ancestry DNA matches, I sent out about 20 Ancestry messages. I did hear back from about 8 of them. In my messages out, I tell them what surnames are in my family, the towns we come from in the old country, and the towns/cities that my forebears moved to in the US. I also tell them about my public tree on Ancestry.
Of the 8 who answered back, only two actually had good enough trees that they and I could figure out how we were related. And for 4-5 who did not answer back, I was still able to glean enough info from the trees that they had posted, to figure out the family connections.
I have also twice sent out real letters, to persons that I was especially interested in hearing from. I included all the above info, plus to whet the recipient's appetite, a copy of an interesting document, like a Russian/Hebrew marriage record. That has worked well.
The only downside - I am now being honored with more family lore tales, invariably of how our forebears are descended from rabbinical families, Sephardic Jews, or fantastically wealthy factory owners who married Russian nobility and moved to St. Petersburg (and these are just the ones I've heard in the past two months). There is an inexhaustible supply of these tales.
I do not approach people more than once or twice, especially persons who I do not know at all. Most people who take DNA tests are not interested in genealogy and finding distant relatives, they just want to know their ethnicity. Plus, I've been annoyed by people on Ancestry who send me messages of the sort: We are 4-6th cousins. Can you tell me how we are related? Invariably these are persons who have no family trees, have surnames like Levy and Goldberg, and expect me to do all the research. No thanks.
Lock/Lak/Lok and Kalon in Zagare/Joniskis/Gruzdziai, Lithuania
Trisinsky/Trushinsky/Sturisky and Leybman in Dotnuva, Lithuania
Olitsky in Alytus, Suwalki, Poland/Lithuania
Gutman/Goodman in Czestochowa, Poland
Lavine/Lewin/Levin in Trenton, New Jersey and Lida/Vilna gub., Belarus