Users assigned to a task are responsible for that particular task
They can monitor their tasks from by clicking on "My Timesheet" in the left hand collumn:
or by clicking on the "Tasks" button and then going to the "follow-up" section:
The project manager can also do so by clicking on the "Gantt" button:
To monitor the progress of the task more effectively, you can edit the % of the task they have completed, make a follow-up assessment, look at the progress graphically in the Gantt etc ...