Generally speaking, if your child can't learn much from group lessons, s/he needs a private coach. Private coach can teach something most needed for your child. It often happens when your child reaches 1000 level. It's not easy to find group lessons at chess schools that can boost him/her upward, unless that is a special training workshop for outstanding youngsters.
You need find a private coach if your child needs a lot of undivided attention.
You need a private coach if your child is playing a lot of tournaments. S/he needs help to analyze his/her games, to find weakness, and to plan training. It's not possible at group lessons, and even if it's possible, you should be unwilling to publish all his/her games, because his/her opponents may prepare against him/her.
You need a private coach if your child is preparing for an important tournament.