Whenever a new student starts with me, I like to sit down and have a chat with them about what brings them to my studio and why they want to take private lessons. The overwhelmingly popular response that I get is that they are dissatisfied with their experience with group lessons in the school band setting, whether it be because their peers aren't as motivated as they could be or because the teacher doesn't specialize in the instrument that the student is learning. Either way, the student feels held back.
As a private teacher, I strongly encourage all my students to take part in their school band programs. The skills necessary to be a great ensemble member are different from those needed to be an engaging solo artist, and school band is a great venue to learn those skills.
However, instrumental lessons are most productive in a one-on-one setting, rather than as a group, since each student will have different needs. Every body is shaped a little differently, so figuring out a healthy flute playing position presents a unique challenge to each person who learns the instrument. Developing a mature sound (or any sound at all, for that matter) also requires a good amount of individual attention from a qualified teacher, since the way the flute produces sound is hard to visualize.
So it's important to strike a balance; I want my students to get the personalized, detailed feedback they need from me to improve as musicians, but I also like them to participate in band since it encourages them to think beyond themselves. Group lessons in band should function as a way for students to figure out if they like the instrument enough to pursue it, not as the sole means of education for a beginning flutist.