The only way non-Owners of Dashboards become aware of a Dashboard being republished is if (a) they happen to notice changes in the Dashboard design since the last time they viewed the Dashboard, or (...
I like dougnewton's idea for native Sisense users. We're embedding Sisense as part of our app, so we'd need a way to display the notification while viewing the Dashboard (or expose such an indicator via the API so that we could add the flag in our own UI when displaying a list of available Dashboards within our application).