Notifications of the task being completed would normally be sent out to all involved in the project, including the assignee (the person who created the task).

It’s important (as usual) for the assignee to ensure – if they are the project owner – that they haven’t opted out of these alerts.

There are a couple of exceptions to this though; for example where a task may have been created by person A and then re-assigned by another.

The second person although reassigning the task is not considered to be the ‘assignee’ since they did not create the task.

They won’t get an alert – unless they are included in the task watchers.