The best way to synchronise multiple JEvents installations is to use the ical import/export facililty. One the master you create an ical export menu item and generate the URL that you will use to import into the slave site - you don't show this menu item on any modules.
On the slave site set up an new calendar in the backend imported from the master site ical export URL. Then setup a cronjob to keep it up to date.
See http://www.jevents.net/en/jevents-15-to ... -calendars
for more info
intranet->public may mean that you can't easily do this since the public site can't access the private master site. If this is the case then setup a cronjob on the intranet site to push the generated ical file to a publically accessible location.
If you really want to use event triggers on an event by event basis there are lots you can use - if you download some of the silver member plugins you can see them in action there.