Ok, this is now implement in latest CB Activity 7.0.0 build release.
You'll now find within CB Activity > Parameters > Notifications > Email you can turn on notify by email. This will send the same contents as the on-site notification as an email to all recipients. By default this is non-personalized so it'll send as a fast BCC email. It can be configured to personalized so every recipient is passed through substitutions, but that will be slow. This can be force enabled for everyone or it can be dependent on a field value to allow per-user toggling of email notifications.
In addition to this I've added the following triggers.
Specific Recipients
Code:
$_PLUGINS->trigger( 'activity_onAddNotificationRecipients', [ $this, $userIds, $fieldId ] );
Relational Recipients
Code:
$_PLUGINS->trigger( 'activity_onAddNotificationRelationRecipients', [ $this, $recipients, $type, $fieldId, $exclude, $userId ] );
These can be used to implement your own recipient processing behavior. For example directly after this is the email handling and it uses information provided by these triggers. So is ideal for things like adding SMS for example if you've an external service to provide that.
As for daily/weekly/monthly digests I've opened a new ticket to explore adding that in a future release as it'll require a new scheduler like feature to handle that.