[Boone Gorges]

The best solution at this time is something like Group Activity Notifications https://wordpress.org/extend/plugins/buddypress-group-activity-stream-subscription/. It does a bit more than what you’re asking for, and by default it has everyone subscribed to everything, but it does allow the user pretty fine-grained control over notifications.

We’re working to make the plugin more customizable, so that the site admin could choose to turn notifications off by default. Then it would work just like the way you describe.