As I said before, chess is a 10-year project. Thinking about your child's school education, how many teachers s/he has? When s/he grows up, s/he will have many many different teachers. It's same for chess. There should be proper teacher for proper time period. So answer to this question is you should and you would.

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.

When we discuss how to find your coach, there was a problem for parents, how can I tell what is the skill level of my coach? Usually a tournament rating should reflect your coach's level, for example, USCF rating. If your coach only boasts about his/her online rating, be aware!

How to look up your coach's rating? You need your coach's name. Go to 
http://www.uschess.org/datapage/player-search.php

Enter your coach's last name and first name, for example (me):
dong,andy

There might be couple of them. Choose the right state. Usually that will be the correct one.

This is the way to find out your coach's rating, in fact, anyone's rating. My rule of thumb is: your coach's rating should be 500 above your child's rating. That's why 1500 is the graduation level under my teaching. It's not that I can't guide anyone above 1500, but rather the student could gain more from an IM/GM coach at that level.

If your coach doesn't have any rating history in USCF, either he/she has retired from chess long time ago, and no longer playing in tournaments, or he/she never had a chance playing in US. You have to evaluate him/her by his students' performance and other parents' recommendations.

If your coach is outside US, you could look up his/her FIDE rating too.

Rating is not everything. Coaching also includes teaching style, curriculum, chemistry with your child, responsiveness outside the class, and whether he/she cares about your child. But rating is an important hard number no one can fake.