Official comment
Thanks for your feature request!
If I'm understanding you correctly, you are talking about the notification emails sent to the task owner? Both types of Tasks only send the Contact Owner and Due Date through on the confirmation / notification emails by default.
You can add the [First Name] and [Last Name] merge fields to the body of the Quick Task or task template to have the contact's information sent through. Otherwise, it is listed with the contact name in Tasks collection.
We'll keep this open to see if there's interest in us sending through the contact name as a default field also.