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 (...
Thanks DRay and HamzaJ -- looking at the options listed, they're all prefixed with "Share" not "Republish". If I were to enable the "Share Dashboard With Existing User" option, would the system email all users (that have had the dashboard shared with them) every time that Dashboard is republished?
Also, follow-up question -- from the snippet, it looks like these are system-wide settings -- is that the case? It's not a per-Dashboard or per-user setting?
-Ben