On each person's profile within the People tab of your Control Panel, there is a tab called 'Notifications.' Clicking it will reveal a menu like the one pictured below, which allows you to specify exactly what that individual person will be notified about via email.

In the image above, I've highlighted the relevant setting to control who is notified when someone is assigned (to them or to someone else, depending on which box(es) are checked); NationBuilder calls assignments "point person changes."