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Both the site admin and the new user should receive mails when the registration form is submitted and when the new user account is activated. You can test the site by registering yourself as a new member using a working email address and see if the script sends you a validation email.
If you don't get the mails, something is wrong with your settings in System Preferences/Mail Setup or User Info Settings. You might need to try PHP(mail)instead of SMTP delivery method.
To check if a new user is activated you can go to System Admin/Edit User, then select the new username and modify. If the account is not activated, XOOPS will display a message page instead of the users account admin.
Hope this helps.