Don J. Modesto wrote:
I say it to whomever I'm training with as we begin; finishing, I say "Arigato Gozaimashita".
Basically, likewise here -- although I usually say, "Thank you!" after I train with someone.
It seems odd to my Japanese ears to hear "onegaishimasu" after training. It's usually said before whatever it is that you wish to have the other person undertake (eg train with you, pick you up at the airport, etc)...
I can say that I've heard senior people say that phrase to juniors, most ostensibly in bowing to start a class when they're teaching it. Of course, it's all intertwined in the Japanese hierarchical society. One might argue by saying, "onegaishimasu" that you're placing yourself "under" the other person (as you are, in a way, asking for a favor of the other party). As such, a person more senior might ask you if you can do something for them by asking, "Can you ------ for me?" rather than putting it as a "favor" of sorts.
But, to answer Ron's original question, I'd say that most of the time, both parties training will say, "onegaishimasu" to each other before training. It doesn't seem odd for me to say such to my partner regardless of their rank...