What is the difference between a coach and a private lesson teacher? Is there one? Should there be one? Can a good coach not be as good at teaching in a private lesson setting? Can a good private teacher not be as good when coaching a group? I do think there is a difference in the two. They are different functions and require their own set of skills.
Some private lesson teachers have not been in a 'serious' (meaning long term playing, working, performing and/or recording together) chamber music situation to know the ins and outs of the circumstance. Some private lesson teachers might not warm to working with groups for whatever reasons. Coaching is dealing with a group entity. Private lesson teaching is dealing with a person with issues no matter how small or big they are. There are similarities of course in these functions but they are certainly not the same.
The group or individual student needs to trust the teacher or coach. The group or the individual student needs to practice on their own and take up what the teacher or coach suggests. A private lesson is a more intimate close range circumstance. However coaching can get pretty close and touchy because of the factor of 'public' exposure or one member being singled out in front of the others. This needs lots of care on the coach's part.
One of the jobs of a coach is to act as a unifying agent or magnet when the group is separating from its intactness. This of course should only be temporary. Unlike an orchestra when all the rehearsals are with a conductor, the chamber group rehearses on their own. This is a real significant difference. The chamber group needs to develop a certain kind of relationship and understanding with each other to be able to communicate and stay on task. The coach needs to be able to spot where the weaknesses are in the group dynamics and make suggestions, offer exercises and/or give techniques for them to work on.
In either scenario, it is a duet between the teacher and student or the coach and the group. Both of these relationships call for agreements, understandings and open communication that is held together by worked out standards and criteria.