ESL teacher training is a common next step for English teachers who wish for a change from teaching regular English classes. This can be a good move, although TEFL/TESL training is still teaching; therefore, it's most suited to teachers who enjoy teaching - of course.
A lot of ESL teacher training takes place 'in-house', in many kinds of educational institutions. Basically anywhere where English is taught there may be teacher training courses to support their teachers. As well as 'in-house' teacher training, there are also many institutes, language schools, and universities which run dedicated teacher training programs. These can be for initial ESL teacher training, leading to TEFL certificates or they can be for experienced teachers looking for higher levels of tefl certification, such as the TEFL diploma. Many universities also have teacher training courses which prepare undergraduates and/or postgraduates to teach English.
Any teacher with any experience in English teaching will have been exposed to different kinds and qualities of training. I've been sat in front of video of a teacher teaching a class of children. I watched it by myself in an empty classroom. After the video had finished, a manager in the school came in and cheerfully told me that my training was finished, and asked when I could begin teaching.
At least in the above case some attempt at training was made. Not all teachers are so lucky, and many teachers have just been given a book and told to go. Sometimes training is listening to a talk about teaching, combined with observations of other English teachers' classes.
Lack of training has become less common as more new English teachers now take some form of TEFL certificate. However, if you're being hired to teach English at university - depending on the country - it's quite possible that a PhD will be considered MUCH more highly than any teacher training qualification. Sadly for the students.
I should mention that it's possible to learn a lot from observations - both from the good and the bad.
In TESOL circles the usual method of teacher training is through workshops and seminars, in which the teacher trainer discusses a teaching point, then says "get into groups and discuss." The 'get into groups and discuss' methodology can be helpful, and is far better than sitting by yourself watching videos, or by listening to a lecture. The advantage is that you have the chance to exchange views with colleagues.
Teaching practice, and feedback on your teaching, is usually included and is very useful - probably the most useful part of most courses.
Observations of experienced teachers are also a part of many training programs.
An effective, but more labor intensive - and therefore expensive - way is to have a trainer and trainee working together, as in the old apprentice system.
The teacher trainer and trainee plan the lessons together, and decide who will teach which part of the class. The trainee might do a warmer with class and then present some new language. The trainer could then teach the next part of the class, then the trainee would observe, then the trainee would teach a bit more, and so on.
This method has worked well for me, and I used it to train teachers for over a decade. It is, however, time consuming, and it can put some pressure on the trainee, although this may be an inevitable part of getting ready to face a class alone. It can also put pressure on the trainer, who will be regularly observed as well and must, therefore, have the necessary experience and ability.
Finding work in ESL/EFL teacher training requires that you have experience teaching English of course. The amount of time required varies a lot deepening on your teaching situation. If all the teachers in your language school are new teachers, and you have been teaching for two years you may be offered a chance at training. In institutions offering TEFL certificate and diploma courses, you will need to have the qualifications you are teaching, together with much more extensive teaching experience.
Most language schools give teacher training work to their most experienced teachers. Often it is the duty of the Assistant Director of Studies, or Director of Studies. If you want to train teachers, first become experienced in different kinds of teaching, and take TEFL certification. You can suggest giving a workshop, or part of one, yourself. This will give you good experience.
The rewards are similar to those of teaching. You can see people who were not confident gain confidence, you can help people become better teachers, you will help their future students have a better experience, and you will have more variety in your work. The rewards can also be financial, as teacher trainers will normally earn more money than regular English teachers. And finally training new teachers can be a lot of fun.