When an unauthenticated visitor accesses a site with email authentication on, he or she will first be routed to the gatekeeper.
The viewer has to enter his / her email address and hit Send Verification Email.
If the email address entered is NOT authorized to view the site, a notification (with helpful tips) will be displayed, as shown below:
You can either re-enter a different email address or click Request access button for the current email. If you click on Request access button, you will get a notification as shown below and a site administrator will be notified through email that you're requesting access for the site and will have the option to allow the access.
If the email address entered is authorized, a success notification will be shown, such as below, and an email will be sent to the associated inbox:
To finally access the site, the viewer can then click the View Secured Site button in the notification email, as highlighted below: