Mark - This is such a complicated question that I need to write a little essay on it alone.
You are my sensei
For a "traditional" student - and this can include a student studying a modern art such as aikido, to become a teacher's student implies a real commitment. This would be particularly true if I was the student of a teacher outside of a headquarters dojo (a modern anomaly of sorts), where a lot of instructors teach classes under the same roof. If, for example, I was Nishio Shoji's student, and I started going elsewhere - to other private dojos - I'd implicitly be saying that I already understood Nishio sensei's information and found someone else a better use of my time. There could be an exception if an instructor was especially respected or a friend. But then, my instructor should make an introduction. For example, Kuroiwa sensei personally brought me to Nishio sensei's dojo. We were late - Nishio sensei stopped class, and he and Kuroiwa sensei enchanged all sorts of pleasantries, and then Nishio sensei walked me back to the locker room to change. This was definitely NOT due to any merit on my part - it was because I was a "limb," so to speak, of Kuroiwa sensei, in this regard.
My Araki-ryu teacher was rather irritated with me when I referred to one of his contemporaries in Araki-ryu as "sensei," saying that I should only refer to HIS direct teachers as sensei. Similarly, I made one to many references to Otake Risuke of TSKSR said, and he looked at me and said, "You seem to be spending a lot of time with Otake-san." I said that I'd visited him twice in regards to my book (Old School). "Well, you sure seem to be talking about him a lot." In other words, why would I be talking about what some other guy - an enemy, in some respects - might be thinking, rather than paying attention to what my own teacher might offer me - unless, I was saying 1) Otake knew these things better than my teacher 2) I didn't see worth in what my teacher might
say, so I was filling the air with something else to talk about. This would apply to a foreign student as well as a Japanese student, except that few foreign students - until recently - were in a position to be regarded as deshi. AND - often some allowances would be made for the mistakes, omissions, even rebellions that a foreign student might make.
My special position
Kuwamori Yasunori understood that I wanted to train with every aikido teacher I could, without it being disrespectful to him. So what he told me was to put Kuwamori Dojo - embroidered - on my hakama. Then, when I went to another teacher's school, the teacher wouldn't be put in a position that a) they didn't know who was I and why was I there. Would teaching me be poaching? In fact, I'd sometimes have an intro letter from Kuwamori. b) Because I was "taken," there was a clear message from Kuwamori that sending me to the other instructor was a mark of respect. I was well-taught. This was an incredible gift to me and one that many teachers wouldn't have the strength of character and ego to accept. It also - I think - was due to the adaptable nature of modern aikido. There are enough similarities amongst various styles (until one goes deep, so to speak) that allow a really focused student to train in various groups. The only way to accomplish this, however, is to drop your own ego at the door. What I did was essentially try to do exactly what each teacher was teaching, and then, back at my home dojo, allow it to sink in to my bones, developing my own personal style. At a certain point, however, I had far less need or interest in going to other teacher's classes, because I was paring down skills to conform to their different styles - or as Bob Dylan says, "You gotta serve somebody."
In fact, Yamaguchi sensei, after my nidan test, which he passed said to someone, "I don't like that fellow all that much, but I have to give him credit. He's way too big to do my style properly - but he's taken what everyone teaches him, and made a creditable method of his own."
What Kuwamori sensei did for me was quite unusual - at least at that time. Most instructors would be more jealous, perhaps, or would have regarded my travels as creating a jack of all trades and a master of none. At Iwama, for example, I don't think such behavior on my part would have been supported.
You shouldn't study with me
I asked Watanabe sensei of Honbu if I could visit his private training dojo and he refused. In essence (and he was kind enough to explain), he said that he liked teaching me, and if I started training with him on a more private basis, it would upset him if I didn't completely commit to him. So he preferred to work with me in the less intense circumstances of Honbu Dojo.
There are more ramifications to this. But there's a start.