Respect and loyalty form the foundation of the instructor/student relationship. It must be mutual and constant. Many things we may not give a second thought to are actually considered to be very disrespectful, and should be avoided.
Earlier tonight, I was contacted by someone who currently trains at another school but was looking to supplement their studies with more realistic self defense training. Upon learning that they were a student elsewhere, I immediately turned them down. If you are a student somewhere, and there is something you have an interest in but are not currently learning, your first responsibility is to talk to your current instructor. Often times, we can accommodate you. I certainly do not teach everything I know within my main program, and many instructors are the same. Perhaps what you're looking to learn is also on the minds of other students in the school, and this "want" can influence your instructor to add an additional class/program, or even offer private training.
If you were to go behind your instructor's back and contact another school to start training, it is very disrespectful to your instructor and in my school would be grounds for dismissal. I am very open, and if a student followed the above protocol I would wish them the best in their training. Heck, if they learned something cool I would ask them to show me. But to not do so is the same as a child sneaking around behind their parent's back. If a child thinks what they're doing is the right thing, why couldn't they tell their parents about it and have to hide?
Another thing to consider is your commitment to your current training. Even if your current instructor approves you visiting or training with another school, your primary art should come first. You need to be absolutely certain that the training schedules do not conflict. You also need to make sure that what you're studying does not teach something directly contradictory to your primary art, because it will develop bad training habits. Some arts are just not compatible, and should be learned separately.
If you have a student who is hungry for the arts, do not try to control them or forbid them from growing their wings. This stems from insecurity and fear. If you're not comfortable enough with your program that you're afraid to lose students if they experience something you can't teach, then you shouldn't be an instructor anyway. If they merely want to supplement their training, you should encourage them. If you've trained them correctly, they will always come back and be your student. We are a martial arts community, and all should be working together to provide the best overall experience for our students. Our number one priority as instructors is to provide for the needs of our students, even if that means letting them train somewhere else part-time or sending them to another school that better fits what they're looking for. Do you really want to train someone who is not happy with your program anyway?