Private flute lessons are an investment, and you should take the time to make sure your teacher is a good match for your needs. There are lots of factors that make each teacher different and can make one teacher a better choice for you. When you look for a flute teacher, consider the following:
A degree is not a guarantee of skill at private teaching, although it is generally preferable to have a private teacher who has a college-level music education. Just be aware that there is no degree program designed specifically for teaching private lessons; even a degree specifically in music education is only meant to prepare someone to teach general music classes in a school, and doesn't focus on any one instrument. Ask your potential teacher if they have studied flute pedagogy, which is how to teach the flute in particular. I was lucky enough that both my undergraduate and graduate programs offered pedagogy classes, which helped me refine my own teaching skills and increased my confidence as a teacher.
A teacher who never went to music school can still be a good teacher, but it will take them longer to accumulate all the knowledge one would be exposed to in music school. Also, if you're looking to audition for music school in the future, it's best to study with someone who has done so themselves. My first private teacher did not have a degree, but she was an excellent teacher; however, when I told her in my junior year of high school that I wanted to go to music school, she told me that I needed to go study with someone who had been there. This was a generous move on her part that I remember gratefully to this day.
How many years has your potential mentor been teaching? Have they been doing it consistently or off and on? You want a teacher with at least a few years of teaching under their belt, unless you're looking to reduce cost; a beginning teacher is less expensive than a more established one. If you live near a music school, ask if they have a pedagogy program that needs students for their budding teachers to teach; you can often receive a month or two of lessons at little to no cost, which is good for children who are just starting out on their instruments. The trade-off, of course, is that a less experienced teacher has had less time to refine their technique and gather resources. The ideal teacher has many different ways to explain the same concept, and finding those ways takes a few years of trial and error.
Some teachers like to focus on one age or ability level. With the huge number of flute teachers available, you can absolutely find one who specializes in your skill level. Ask your potential teacher what ages they teach most often and what ability level they prefer to teach. Make sure that your needs will be met.
You will find that each teacher you meet has a different teaching style. Pay attention when you first sit down to speak with a potential teacher; their demeanor will be a good indicator of how they teach. If you are an experiential learner (someone who learns by doing), find a teacher who demonstrates but then invites you to try a new technique. If you need a new idea explained to you verbally for you to understand, find someone who is good at talking things through in a clear way. If you need tough love, there are plenty of teachers who take no excuses during lessons. You might not find the right fit right away, and that's ok. It's worth it to keep looking until you do find it, because the progress you will see is amazing.
Private lessons are an investment. Compare pricing for a few different studios to get a sense of what the norm is in your area of the country. For example, in the Boston area a half-hour lesson tends to cost somewhere between $20-35. As mentioned above, a teacher with less education or experience will charge you less than someone who has been teaching for decades, but choosing to pay more can mean a better, more productive private lesson experience.
How close is the studio to your home? Is is convenient enough that you will go to lessons consistently, or will it become unmanageable? Figure out how much of a commute you're willing to make; lessons are only beneficial when they happen regularly, so make sure the trip to get to the flute studio is one you can do weekly. Also, consider the neighborhood where the teacher's studio is located; make sure it's one where you feel safe and comfortable. Some teachers also offer in-home lessons; this is another option for extremely busy people who don't have the time to travel to a music studio.
Whether you're looking for a new teacher for yourself or for your child, don't be afraid to be inquisitive when you interview potential teachers. The more you know, the better of a match you can find between their teaching style and your learning style.