The Private Form Access Email is sent when:
When the visitor to the page first visits the page, they see a message such as this one:
In other words, the Private Form Access Email is sent to confirm that the person that fills out the form is indeed the person that has access to the email account specified.
You can edit this email message by selecting Profile Updates from the Marketing+ menu and then clicking on Private Form Access Email. The message is sent in the scenario described above.