Currently, this cannot be done within Tableau. However, this is a good one for the Ideas section!
Metric Insights' Push Intelligence for Tableau is the only tool I have found so far that will do this.
Just for posterity..while this is still currently possibly only with third-party tools, kettan has compiled a great list of options for this here:
Since this post was authored, we built and started using our own tool, VizAlerts, that we are using at Tableau. It's open-source and free to use.