Within Teamwork, you can choose to notify certain users when a specific task is completed by adding them as followers of the task.
You can add followers when creating a new task or editing an existing task 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.
Use the checkboxes on the left to select the relevant users or companies to follow the task. Once selected, click the button to the right of a user to customize which tasks updates they will receive. You can also go to the Teams tab and set follower preferences for an entire team.
You can also adjust your profile preferences to receive task completion 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. You can toggle the Receive task/milestone completed emails on or off.
For more information, see: Adding Followers to Project Items