I understand that 'community' is a bot https://webmasters.stackexchange.com/users/-1/community but I am curious if each SE site has some control over how it works specifically for that site.
3This link has a wealth of info about the community user: meta.stackexchange.com/q/279203 and this has some info about how it is slightly customized per site meta.stackexchange.com/q/48578– Stephen Ostermiller on Strike ModFeb 5, 2017 at 11:28
This post explains who is the Community user, which works in a consistent way across all Stack Exchange sites.
Users and moderators do not have direct control over how it works, though we can affect when it works, such as by voting to delete a post or marking it as spam (after a number of votes, the Community user will take the corresponding action).
So the short answer is "no", each site doesn't have any control over how the Community user works, but if it does something odd, please post a question about here on the Meta site with the "support" and "bug" tags so the Community Team will see it.