When an new activity is assigned to a user... they will only see this if they go to activities and look for it... it would be very helpful in speeding up response time if the 'Activity Tab' turned a different colour (RED) and maybe even indicated the number of unread activities assigned to the user.
Seems like there are two types of notifications:
1. New activity created; may be due today or next year
2. Activity due in the next X hours or minutes
So really this should be user-defined alert, otherwise there may be a barrage of unwanted messages…