My son has been going to a private chess tutor for about 3 years. We live in the Boston area, and the rates here vary from $40-$75 per hour.

For your first chess coach, the most important thing is to find someone who has a history of success with teaching children, not the one who has the highest chess rating. I recommend staying away from GMs as the first coach, unless they are absolutely excellent with children.

Instead, meet the coaches whose students have won the state championship and have done well nationally. And among those find select one that you think is the best match for your child, and of course, one you can afford. You want someone who makes chess fun but can also facilitate learning. I recommend at least 500 rating points difference between the student and the coach. Your child's first coach should be rated about 1800 minimum. If your coach is too far below that, they will likely teach your child some incorrect methods that need to be corrected later.

For son's first coach was an Expert, meaning that has an official chess rating above 2000. After my son approached that rating, we switched to someone higher. If you would like more information, send me a PM.