I don't think there is a specific personality type that you can attach to teaching. Obviously traits like patience, being outgoing and being able to think on your feet/ improvise if needed are essential, but after that it gets a little more unclear. I specialise in teaching under-16s, I think YL teaching takes someone with a lot more energy and an ability to relate to younger people. When teaching adults or older teens, you have to have a different skillset. I've often found adults to be more frustrated learners as they tend to expect to be fluent after just a few lessons! Obviously this is slightly different when talking about university students, as they are used to academia. Adults deciding to take English classes tend to need a more calm and firm personality teaching them. But these are just my thoughts based on years teaching both children and adults.

I would say the most important personality aspect I look for.in a fellow teacher, is someone who teaching because teaching is what they love doing, because providing a good education is.paramount to them, that their students progression is their main.priority. If you are teaching to fund a lifestyle and the advancement of your students is (at best) secondary to you, I don't want to work with you.

This is not to say I don't love my free time, I have an active social life but if I have forward planning to do for an exam class, or I have homework to mark, it will get done (in my free time) before I consider going out.

I guess the personality trait I look for the most in another teacher is dedication.