I support having both local FB pages and a global one. Convert the existing ‘global’ one to AUS and create a new for the global community. There’s no need to create another FB account, that’s just messy. The person creating the page can add others as admins/moderators, so one could imagine a global ‘communication team’ that manages the global page with contributors coming through the Contributors’ Mill 
This of course requires some extra effort with coordinating this page and team so it depends on if we have the resources available.
I also support a global FB group. Please make this Closed and not Public. We have a local group for the legal entity behind OFN Scandinavia, not very active atm but useful to quickly share interesting articles etc.