Yes, a user the uses JIRA Service Desk to submit a request must be included in the license and your JIRA and JIRA Service Desk licenses must match.
Alternatively you could tie JIRA Service Desk into an email account that non-licensed users could send a message to create issues.
See the first comment on https://blogs.atlassian.com/2013/10/introducing-jira-service-desk/ for more information.
As Jeff mentioned above, email import is one way to do this. We have a fictional "External user" account that is designated the reporter if the email handler can't find a corresponding user account for the email. A couple of other options would be to use an issue collector on a website, or just using a form on a web-app that uses the API to create an issue on behalf of a fictional "Form user". You could then collect their username and store it in a custom field, etc.
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