Accomplished musician does not always make an accomplished teacher. Pavarotti was a great singer, but I would not have brought my child within a mile radius of him. Domingo on the other hand, he would be a great teacher of children.
There is more to a good teacher than just the mere mastery of the subject matter, (That is important, though.) to me the natural born teacher can achieve more with people than virtuosos whose egos are so huge that they can barely share a room with a beginner, never mind teach one.
In the end teaching is a selfless act. You are making a sacrifice in time and money to teach young people. There has to be a certain amount of emotional maturity to be a teacher, something which some great musicians simply do not have.
Many times as teacher you have to think further than just what I want out of the situation and think what does the pupil need. This you cannot do if you think the whole world revolves around you.
There is also a great deal of social graces you need as the teacher. Working with children is not easy. You need the negotiating skills of an Ambassador to get them right.
So in closing what I would say is more important is does this teacher speak with authority? If he is a man how does he strike you? If he makes you uncomfortable then that should be a warning sign.
What are his qualifications, what results has he achieved with his pupils? Most importantly of all, how does he work with children and adults?