That depends on the school. Typically as an ALT teacher you'll be advised not to discipline the students (you are just an assistant) and the head teacher will do that. In a private institute you are more likely to be in charge and not working with a co-teacher for each of your lessons. Whether you have to discipline or not comes down to the eikaiwa in question.
You can ask them.
As an ALT teacher your main role is often to plan lessons and add fun to the lesson. You may find that some students get away with a lot.
If you like independence and autonomy in your work (teaching) then working in an eikaiwa might be best.
Learn more about teaching in Japan.