Well, of course our breathing patterns and our breath in general reflects much more than what/who we are as a musician (if one does not believe that musicianship itself is all of who we are), it is a sign of our very life. Most of what I think about is not magic nor hidden knowledge, but simply what I have observed in myself and in others and a lot of it is common knowledge, actually.
There are a couple of parts to it that I am paying attention to, one being how a person's breath reflects their overall Psyche -- in general, and at the time they are with me. Yes, some of that has to do with nerves in the moment, but some of it may have to do with whether they are a nervous person in general. This alone will tell me a lot about how sensitive I need to be with them as we interact through our lessons together.
This aspect of their psyche will have an effect on their musical expression, but in mostly sub-conscious ways. For example, a "nervous person" (with breathing (or holding breath) to match it) may have a very difficult time with silence and therefore have a difficult time wanting to hear silence within rests and at the very end of a piece. They will often play right through rests and/or at the very end of the piece (especially if it is actually a slower and more "soothing" piece of music), funnily enough, they will resist holding out the notes at the end and hurry themselves to a silence, at which point they will become very uncomfortable (creating in them a reactional seeming-need to nervously
move some part of their body in "self-defense," destroying the experience for themselves and/or the listener -- no matter how beautiful the performance was). I expect almost all of my students to perform on some level (I don't actually require it, but I expect that they will naturally *want* to after they see how wonderful it is to play the piano
), at least I expect them to be thinking artistically from the first note to the last -- so this matters.
The other aspect is the very fact that music has its origins in breath (and some kind of pulse), and that which we may do on a more conscious level. In terms of phrasing, for example, breath is going to play a pretty important role (some people may be more of a "natural" at phrasing then others, and this can be tied to their breathing patterns and awareness of their own breath). Also, proper breath-support plays a huge role in posture/tension/physical motion (and mobility), note accuracy, rhythmic timing and so on, all of which does cross over into the first "category" I mentioned (but here we will use breath more consciously to mold our artistic expression).
This is all common knowledge, mostly. However, I am also a singer and perhaps I pay particular attention to this (many people who use their breath in a conscious way (like a person who practices yoga, for example) probably would pay particular attention to this).
Anyway, that's the general idea -- and, of course I don't personally have it down to a science. One of my favorite thinkers and writers on the subject of life relates breath to inspiration, I think that's interesting.
Topic: First meeting with student