Just like a calender or folder, the task would need to be shared to be visible in the To Do Bar. Even if the user has full access over the others mailbox, Outlook will not set tasks in the the To Do bar automatically.

Simply shared the task(s) you would like to monitor and they should show in the to DO bar once they have been accepted. An Email request will be sent from the owner to the user who the task is being shared.

Best Regards,