private or a public school ? (Hamburg)

Dear All,

Do most working parents prefer to have their kids in a private school or it's very usual to simply go to the school which is next door to your home?

We are a family of musicians and will arrive in Hamburg in Protected content ! I am struggling with finding a grundschule for our two kids and would very much appriciate your advices!
I was wondering what is the mantality in the city about regestering the kids in schools?

If you have kids, let me know how you decided where to register your kids :)

Greetings,
Kristina