Is it permissible for a Jew to eat meat that isn't treif or forbidden flesh if it was not purchased from a kosher butchery? I was told by a Muslim friend of mine that non-halaal meat (except pork) isn't haraam (forbidden) even if it wasn't slaughtered with a prayer as they do. Is this true in Judaism as well?
No. A kosher animal must be specifically slaughtered and prepared according to Jewish laws in order for its meat to be kosher.
These laws are very specific, governing the knife used, the method and stroke of the knife, the method for soaking/salting properly, and checking the innards for defects which would render the animal unkosher. If any of these laws are not followed, then the meat is not kosher.
As such, Jews will only buy meat from an individual/store/company that they trust to follow these laws.
Kudos to both answers above; just one more point: after the ainmal is slaughtered by a Jew according to Jewish law, the meat is then inspected, soaked, and salted; then it can be packaged in a reasonably tamper-proof container and shipped off and sold at any general supermarket. So you don't have to go to a "kosher butcher shop" per se.
Hod's answer above is correct. However, to address your point about "a prayer":
One must recite a b'rachah upon slaughtering an animal, which is the typical practice when performing mitzvos (Rambam, Hil. B'rachos 11:15). However, as usual with birkas hamitzvos, the absence of this blessing does not render the slaughtering unfit (Bi'ur HaGra, YD 1:31).
Eldad Ha-Dani, in his account of the religious practices of his claimed Jewish tribe in Africa, wrote that meat slaughtered without a b'rachah is forbidden. However, as the linked Wikipedia article notes, a number of Rishonim have considered his account unreliable (e.g. Ibn Ezra, Commentary to Sh'mos 2:22). Regardless, Eldad's extra-Talmudic requirements in hilchos sh'chitah are not accepted in practice (Bi'ur HaGra, YD 1:30).
To be clear, it is not the blessing that primarily characterizes Jewish slaughter; it is rather the careful attention to the many detailed and subtle laws governing the slaughter of kosher meat, as well as the fact the the slaughterer must be a God fearing and halachically observant Jew.