Re: My Sensei can beat up your Sensei
When a student finds herself in this position--a guest at a dojo whose sensei is getting hung up on "mine is better than yours"--perhaps she can contribute in a positive way by just not playing along. "I came to learn what you have to teach, not to worry about who is better or worse" followed by resolutely not making comparisons. That's what I've tried to do when I'm a guest. Usually the uncomfortable moments are in the first class or two, and after that if you refuse to get political things simmer down.
The most awkward moment I've ever had along these lines involved a sensei who gave a brief and rather slanted account of the difference between their tradition and mine, in front of both me and one of their own junior students. The junior student said, "That bit you just described, I think that's something I'd really benefit from learning" and the sensei turned an interesting color of pink and changed the subject. I probably turned an interesting color of pink too, because I really felt put on the spot. But I've remained on good terms with that sensei and their school, so I guess I handled it okay.
I think students can contribute positively by not allowing themselves to be drawn into this kind of unhelpful dogfight. If it doesn't impress the students, the teachers are less likely to do it....