If you want to keep track of what your users are doing daily then you would want to Subscribe to projects. If someone is assigned or subscribed to a project then they will receive email alerts anytime another user makes a time, expense, note, or file attachment into the program. You can also have a list of all Assigned/Subscribed Projects on your Dashboard which makes it easier to access them quickly.
If you want to be assigned/subscribed to projects but do not want to receive alerts then hover over your name at the top right and click on My Profile. Then click on your Email Alerts tab. From there uncheck both Assigned and Subscribed boxes and click Save.
Also if you decide to Unsubscribe to a Project then open the Project and click Unsubscribe. This does not prevent you from being able to enter time to this project.