After quite a lot of discussion, we came to the conclusion that it would be easiest to say that Q extended a universal translation field around the Enterprise itself which serves the same purpose, without the crew having to give out some very advanced technology to their new guests. So for now, the abductees don't have access to the commbadges, but when we get to doing Away Missions and such we will figure out a more portable and less-reliant-on-Q solution.

Does that help answer your question?