The notion of private schooling in France is very different from that in, for example, the UK. The first thing to remember is that, in France, private school does not equal better school so you will not be making this choice in order to give your child a 'better' start in life or a 'better' education.
Indeed, a big plus of the state education system in France is that the brightest/middle class/better off pupils are not creamed off by the private sector leaving the 'rest' in the local state school. Tomorrow's car mechanics, lawyers, dentists and farmers rub shoulders in the classroom at least until lycée.
So long as the state school is working well for your child, they will probably get a better education there than in a private school Nevertheless, there are reasons why you might want to go private, generally either because you want your child to have a religious education or, probably more commonly these days, because you feel that your local state-school is not right for your child.
Perhaps it is too big or you feel your child needs better discipline or, for whatever reason, they are simply not getting on well there.
In general, private schools are smaller and class sizes are smaller. And they tend to be slightly less 'anonymous' - involving the parents more in school life.
Another, increasingly common reason why working parents, especially, are opting for private schools is that teachers are less likely to go on strike - at least far less often than those in state schools.
Most, although not all, private schools are Catholic and one of the reasons the French may opt to send their children to a private school is so they get a religious education. However, fear not, Catholicism - indeed, religion in general, is not forced on the children. It is more a matter of a few religious icons, such as crucifix, on the walls and a Christian pastoral approach to children. Catechism classes are optional.
Another big difference between going private in the UK and in France is the cost! Private education here is not at all expensive - expect to pay around €20-40 per month as well as slightly more for lunches than you would in a state school. So long as the private school is registered within the education system (and most are) you will still be entitled to all the same grants and benefits as you would were you in the state sector.