If "they" are Peter and Tom, and Peter is arguing with Tom, then say "They were arguing". That's it. You could say "They were arguing with each other" but the last three words are unnecessary. In that context, "both" is redundant.
However, if "they" are Peter and Tom, and Peter is arguing with Helen and Tom was arguing with Nigel, you could say "They were both arguing" but you'd have to go on and explain who was arguing with who. Note that the word order I have used is much more natural than that in post #1 - I won't go so far as to say you won't hear "They both were ...", but it's not common.