You can receive notifications about tasks you are not assigned to by using the Follow feature. The task will not appear in your MY TASKS or Home page Tasks view but you can get updates on it and view it in the space.
On the Tasks tab within a space, hover over the task name until you see a "Follow" icon to its far right. Then click on "Follow" to add yourself as a Follower.
If you are a space owner, creator of a task or someone in the Assigned To field, click on the task you wish to receive notifications on to open the task details pane. Click the people icon to the left of the task name and use the “Followers to notify” box to add yourself as a Follower. Then hit "Save."
Note: If the task is currently assigned to Everyone, you will need to de-select the Everyone group before you can sort people into Attendees and Followers.