There's a field on sys_filter that specifies the user (it does not appear by default). If it is blank, it becomes a global filter (out-of-box global filters include Active, Active - Unassigned, etc. for Incidents and similar tables). There are also group and domain fields (again, not appearing by default) that do the same, so that if there's a filter that's useful for a group, they can all have access to it.
The "Edit personal filters" link takes you to a filtered view of the sys_filter table that specifies "User = [current user]". And then when you create a new filter from that view, it automatically applies the current user's name.
I hope that makes it more clear. We should document the saved filters better.
