Well, to be fair, it's always hard to come up with a good reason why two Super Sentai teams would want to fight each other. The best excuse was probably in Shinkenger vs Goseiger, because the only senshi they fought was Shinken Red due to the fact that he was brainwashed, so there was no plot hole or leap of logics right there.
Well, yeah, of what I've seen of the Vs. movies, there's usually some brainwashing involved (like Gao vs. Hurri) or some sort of fake-out... the only other versus movie I can think of that involved the teams fighting in good faith was Car vs. Oh, and in that case it does rely diagetically on the Carrangers being idiots. :laugh:
(That's only from what I've seen and what I can remember, though; for some reason I can never actually remember anything about Mega vs. Ginga, even though I watched it pretty recently. Someone more well-versed in Vs. movies could help me out.)
Anyway, I think the reason the Goseigers wanted to steal back the keys so badly is because, again, they had no reason to think the Gokaigers were going to use them to protect Earth. Especially since the Gokaigers, uh, come right out and say, "we don't care about protecting Earth." Despite the fact that they've already demonstrated that they do, repeatedly.
The Goseigers should know this and the Gokaigers should be able to admit to this. But they don't, because then the misunderstanding would be easily resolved and the teams wouldn't have a reason to fight. That's why it's Idiot Plot.