Right now there are only type of people of a task:
Task followers: get notification if something changes in the task
Task assignee: gets notifications as followers plus when some task is nearly due
There should be another role which is assigner (or supervisor / boss, etc.), which is more closely related to how employee interactions are in real life.
This boss would get notifications when his subordinates tasks are nearly due.
He can get a view of all of the tasks and due dates of only his sub-ordinates
Ideally, he can customize the notifications (like notify xx days before tasks are due, etc.), but these are not specifically for task assigners but all users in general.