When your kid just starts, the teacher should be able to interact with kids, and stimulate their interests. Any after-school chess clubs or library lessons should be fine. At this moment, teachers' chess skill is not important. It's more about environment and companions.
When s/he gets interested in chess and wants to learn more, it's the time to find a teacher who can help build up his/her foundation well. Proper training at this stage is important for the long run. If you just skim on it, your child will know a lot of chess terms, or knowledge, but won't be able to use them. It will be very disappointed when they get defeated again and again in tournaments. A lot of exercises is required. There is no shortcut. We, chess schools, actually help provide that kind of training. Our Steps curriculum are requiring students to go over thousands of tactics training step by step.
After s/he gets more mature, and comfortable with tournaments, usually it's after 3, or 4 years of basic training, s/he will need more professional, specialized training. At this moment, s/he will need a coach at IM/GM level. Most times the lesson will be private, specially for him/her. Even at this stage, you may want to change coach once in a while, maybe a year, maybe two years, because different coaches are strong in different areas.
So overall, your child may have 5 or more coaches in his/her long chess growth period.