By default all the site members will receive the record level notifications (add, edit, deleted, record assignment, approval, etc.). You can change this behaviour by settings in Notification Settings section.
Go To Site Admin (It is located on bottom part of Site menu or on the top left corner of the Site)
Click “Notification Settings” link in left hand menu
Click “Update” button located on top right corner of page
Notification Settings can have any one of following values
Enabled: This is default. This has got two possible options
All Members: This is default. This implies that email notifications will be sent to all the members who are having access to app and/or record.
Selected Members: This implies that email notifications will be only sent to selected users. As soon as you click on “Selected Members”, a section will appear below which allows you to select site members. The section shows all the Site Members already selected. Click “Browse – Site Members” button to select new Site Members. Select existing Site Members by clicking onto the corresponding check boxes. Click “Apply” button.
Disabled: This will disable all the email notifications