Living DNA and GPS Origins do not give you lists of matches. Living DNA does provide your "motherline" haplogroup (that is, your mitochondrial DNA) but only a very broad label (such as H), and for men the "fatherline" or Y-DNA haplogroup. They gave me a percentage of how much "European" ancestry I have (but not further subdivided).
GPS Origins gave me 14 regions where I supposedly have ancestry from--some very unbelievable and not at all consistent with what I know from genealogical research.
23andMe gives mitochondrial and Y-DNA haplogroups (but rather broad categories). MyHeritage and Ancestry don't give you those. FamilyTreeDNA does if you pay extra for those tests (and their categories are more precise).
Having done all those tests, I have tons of matches who are somehow distantly related but often I don't know how. Usually I can tell if it is on my father's side or my mother's side. Sometimes I can figure out from seeing what other matches that person has--but in many cases that doesn't help because none of the other matches have surnames I have seen before.
I have had some interesting exchanges with distant relatives whom I would not know of if we had not both taken the test and have been able to help others with information they did not have.
I'm not worried that any close relative might commit a crime and get caught because I've taken a DNA test.
Thanks for all the great info. My father was born in Holland, and my mother in Canada, but her family is originally from England, and settled in the U.S., and there’s supposed to be Mohawk blood on her great-grandmother’s line. It will be interesting to see if any of that shows up on a DNA test, and where the European roots connect to.