we had this similar issue and we had to enable our users to have access to the specific inbox that the system uses to send this particular notification (it works differently to the email update notifications), we then gave access to this inbox to our users (it does show in the drop down box in the email action button, we have raised this to be removed by Hornbill, we had to set the default inbox for each service so Hornbill doesn't automatically use this to send standard emails from the system) but we now get an email telling us when a request is assigned to us.
The role needs to be attached to anyone who assigns requests, not the people who want it, so we gave it to everyone in case