Currently, each site of the SE network handles user issues separately. Moderators don't have access to the suspension history and annotations on other SE sites, they will generally judge users based solely on the history the user has on their site, previous behaviour on other sites doesn't matter.
In some cases a moderator might have heard about a specific users' actions on other sites as they were discussed in the moderator chat room. In those cases the moderator might be influenced by this knowledge and therefore less inclined to give that user the benefit of the doubt he might give to completely unknown users. But the moderator actions are still solely based on the actions on that specific site, not on the history of the user on other sites.
Should the history of a user on other sites matter for moderator decisions, or should we keep moderator decisions on the site level, and ignore the network-level behaviour of a user for local decisions?
To make it more specific, should a user that has been suspended multiple times on other SE sites be treated the same as a user that has no history on other sites? Or should a negative history on other sites lead to quicker suspensions for problematic behaviour or longer suspension times than for users without a history of annotations or suspensions?