I have a react native project that I resisted using flux (in my case, mobx) as long as possible to try this theory out. The situation I ran into that instantly told me I needed global state management was when I had an index screen for list of resources, and a edit screen for a single resource. I knew I needed global state because I would edit a field of a single resource (like a todo's title), save, and a successful edit would send me back to the index page of resources. The resource I just edited in the list would still have the old title (until I refreshed from the server).
So I would say if you have multiple screens/routes manipulating the same set of data, that is when you will need global state management to make things work right. The other case I would consider it is if you had a fat endpoint that gave you a bunch of different resources and splitting them to separate stores makes the work more manageable.
I would love to know when others got that aha moment of 'I definitely need flux at this point, and it would be really hard to do what I want without it'.