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 (...
Rather than email, could this be done in-app?
For example, when looking at the cards of dashboards:
There's no indicator that something has changed recently, unless you hover over the card. Maybe some sort of icon could be added here, to show that it was changed since the last login.
Also, there's an alarm-bell in the upper right for notifications. Could changes to dashboards that I have access to, be listed there as well?