can I use a theme on user sites with child theme?

I am using a child theme on my main domain site, but now want to use the same theme for a user site (with the child theme activated). My question is: is the child theme on each site unique to that site or are they the same? In other words, if I update the child theme on the network, will it over-write both child themes?

I see that I can only rename the child theme while I'm logged in as the super-admin, but not edit the child theme logged in as a user.

Does the child theme method work with multisites where you want to use the theme on different user sites?

Thanks, Joe.