Hi Katie,
Kristen beat me to it, I've done exactly what she mentioned above which caused me to receive the email. The alert campaign that I set up has the lead owner (me) and the send the alert to: with my email address. If that's the case, change the send to: (none) if the lead owner is selected, if you don't want the lead owner to receive the email.
If the form you're using is on a couple of landing pages, but if you don't specify in your smart campaign that delivers the alert email which landing page, you'll get a couple of alert emails for sure for the form fill. So do:
fills out form: the form name, add a constraint by naming the specific landing page for which you want to receive an alert from.