Can I set up alert notifications for the Assigned to field in emails?

You can set up alert notifications for the Assigned to field in your emails. So if a contact or an organization sends an email, the email recipient and the email record owner will receive a notification. 
This new configuration is available In Actions > Engagements Alerts > Email Alerts as Contact sent an email.
The record owner or a contact assigned will be notified via desktop notification whenever a contact sends a new email. (Task, Ref)
Note: If a CRM user disables the alert (engagement alerts are set at a user level), then the contact will not be notified about the email.