RE: sure where messages sent from user to another are sent to.
Messages are set to the entry’s preferred email address. If a preferred is not set and the entry has more than one email address, it is sent to the first email address. If the entry has a single email address, it is sent to that address. If there are no email addresses, the contact form is not shown.
RE: There doesnt seem to be a spot in WordPress’ back end to receive messages, nor was a message sent via email to the user.
Email is sent via email, not to an area in the admin. The user should check their spam/junk folder.
You can also check check if the email is actually being sent. Important fact, Connections, nor does the add on Contact, send the email. WordPress sends the email.
Logs of the email can be found on the Connections : Tools admin page under the Logs tab. The Sent status will show if WordPress reported success or failure. What WordPress reports is based on what the web host reports to WordPress.
I suggest installing the Post SMTP Mailer/Email Log. No need to make any configurations changes to it after install. Send a test message thru Contact and then look for it in the Post SMTP Email log. You can view more details about the about the sending process by looking at the email transcript.
If everything is reporting the email as sent successfully and the email is not is the users junk/spam folder then is was likely blocked during transport. There can be many reasons for this. The first thing I would check is that your sites IP address is not on a spam black list. If you are on a shared host, this is somewhat common as another user on the same IP address as your account could have had the IP address blacklisted for sending spam.