The whole teacher/student thing is interesting to watch. At a big residential place like San Francisco Zen Center (SFZC) where I've been staying for the past week or so, it's vital to keep some clear lines of demarcation between teachers and students. The rakusus (little bib-like thingies you get when you receive the Buddhist precepts) are color coded so that you instantly know who here is a priest, who is a lay person and who has dharma transmission. Blue is for lay people, black for non-transmitted priests and brown is for transmitted teachers. Only priests can wear an o-kesa (the sash-like thingy over your shoulder), though other people can wear robes. And non-black o-kesas are a big no-no for anyone not having received transmission. There are written rules for how these various levels should interact. If you want to keep a large scale residential practice center operating smoothly, it's nearly impossible to avoid some variation on this kind of practice.
It's very different, though, from the way I came through the ranks. Even in the matter of rakusus, Nishijima Sensei's policy has always been that whatever color you like is fine. In his lineage anyone can take the precepts, and anyone who does so he considers to be a monk. There's no division between a lay person and a priest. Everyone is encouraged to wear o-kesas, though even I seldom do (sorry, sensei!). Dharma transmission is a different matter. You don't just get that cuz you ask for it as is the case with the precepts. But this too is handled more loosely than it is at SFZC. In Nishijima's line there are no specific steps you must complete before being eligible for transmission. It's more a matter of how Nishijima Sensei or one of his dharma heirs feels about your ability to understand the teaching and practice.
In the worldwide Zen community you'll find all sorts of variations from super tight Soto-shu style which makes even SFZC's style look like a cakewalk, to super loose teachers who make Nishijima's style look like pure Japanese authoritarianism.
In my own case I have been avoiding calling anyone who studies with me "my student." I remember dealing with one Zen teacher who, the very minute he got his robes and started teaching, was all "my student" this and "my student" that. He seemed to cram those words into everything he said just to make sure you knew he had students. It was comical, I tell ya! Maybe that's where my aversion comes from. But casting my mind back to days gone by I'm hard pressed to think of a single instance when my first teacher, Tim, ever called anyone his student. And I can only think of a few times I've heard Nishijima Sensei say it either -- though he does not avoid the word entirely like I do.
The usual teacher/student relationship involves authority and power. As such it can easily be perverted. In fact it seems that it requires great care just to keep such relationships from getting weird right from the get-go.
Anyway. Even once I start giving the precepts to people I will still won't consider any of them "my students." They'll still just be my zazen buddies, like some people have drinking buddies or golf buddies. Please do me a favor and don't start thinking of yourselves as my students either. OK? We cool? Good.