Another after thought is that there can be of some valuable networking from whom you take lessons. I wouldn't let that be my main reason for choosing a teacher but it can be a spin off.
For example, through my H.S. and college trumpet teacher, I got recommended to play in the Royal Hawaiian Band. Through my first alto sax teacher, a reference to one of the best composers in Japan. Through another sax teacher, I got picked to play Lead Alto and then Solo Tenor in one of the workingist jazz bands in the region, and from my composition teacher a reference to one of the best composition jobs in the country.
In each case, it paid off financially and professionally. But that's not why I picked any of them.