[quote=Don J. Modesto;77199]I say it to whomever I'm training with as we begin; finishing, I say "Arigato Gozaimashita".[/quote}
That's the actual correct form.
Don J. Modesto
I've noticed a trend for saying "Onegai Shimasu" both before and after training. I have a couple of students from an Iwama dojo who do that and also say it to me after I've offered a suggestion on their technique. (I'd be interested in comments from people who do this.) I always say thank you in that situation myself.
In Japan, it's used whenever a request is being made and then having been made and granted, used again, i.e., they don't say thank you at that point as we do in English, they repeat "Onegai Shimasu" (in the logic of Jpn protocol, thanking someone at this point is presumptuous somehow).
That, I think, is a misunderstanding.
It's not generally considered presumptuous to thank someone after they've granted your request, but Japanese speakers may sometimes say "onegaishimasu" again, especially if the one who helped them threw in a comment or two that would enhance the thing they did for them. In that case, it comes closer to the meaning of "yoroshiku onegaishimasu," which is sort of like saying, "Please keep looking out for me," or "Please keep doing this sort of good thing for me."
Non-native speakers can easily miss the thanks among the flurry of words that might surround any exchange between Japanese. But they might note hearing "onegaishimasu" again and get the idea that you say it both before and after you get what you want.
A little further on that, "Onegaishimasu," literally means "I am making a humble request," or "I am asking for your help."
In casual settings, it can be shortened to "onegai," usually after a request has been made a couple of times already. In that case, it translates more or less to "I'm asking you!" and is rather insistent. It's not likely to be heard in the dojo.
But I think it's important for aikidoka to understand the everyday meaning of words to appreciate the deeper meanings of "martial" words. Thinking the two can be separated loses a lot of depth in meaning and leads to people approaching aikido as a kind of cult.