How do you delete a remote git branch “properly”, a.k.a. updating the remote branch list for all users?

Until they update their remotes, their git will have no knowledge of what’s happened on the repository. Once they’ve done an update (via git fetch or git remote update), git remote show origin will correctly show that they have local tracking branches for branches that no longer exist upstream. At that point, git remote prune can be used to remove the stale local branches.