Within Teamwork Projects, you can choose to notify certain users when a specific task is completed, by adding them as a follower to the task.
You can add followers when creating a task or update existing followers by selecting the 'eye' icon to the right of the task name. You can then select who should follow and choose what notifications each follower should receive.
Select 'only task changes' for the user to receive updates only for when the task is marked as complete.
You can also adjust your profile preferences to receive Task Completed emails. When a task is completed on a project you can receive an email to notify you. You will receive emails for any tasks you have completed and/or task which you are assigned to that have been completed.
Click on your profile icon to the top right of of your site, and select 'Edit My Details' from the dropdown menu:
Select the Preferences tab in the pop up modal. Here, you can toggle the 'Receive task/milestone completed emails' on or off:
For more information see: Adding followers to various project items