I have to admit it bothers me how logically inconsistent this is. The word "each" on its own is in reference to a multitude (for example "he spoke to each of them"), whereas obviously the words "one" and "an" are singular in nature. Why, then, should this meaning logically flip around when adding the word "other" onto the end?

@SteveMelnikoff While this may be true coloquially, just because such phrases are used in practice doesn't make them "correct". I would argue that "one another" is indeed more correct in your example.