Re: Two teachers...
My students are certainly welocme to trainwith whomever they wish. I trained for quite a while with both Mary Heiny Sensei and Bruce Bookman Sensei while maintaining my relationship with Saotome Sensei as my teacher. I ran a dojo under Chiba Sensei's within the USAF Western region while being a student of Saotome Sensei and a personal memeber of the ASU. In no circumstance was I ever less than absolutely open with anyone about what I was doing.
Those folks who maintain that it is not the teacher's business whom you train with demean the relationship between the teacher and the student. Anything which has to do with my student's training is of concern to me. They are welcome to train any way they need to but if they lie to me or go behind my back they are clearly not students whom I can trust and I will not invest in them. If you feel that you can't tell your teacher anything about what you are doing with your training then you need to find a new teacher, not dishonor yourself by becoming a "sneak".