The most common workaround to this situation is creating a new permission scheme for your Service Desk project in which you allow the group "Anyone" to create issues and even Browse some if you want.
The only problem would be not being able to identify by internal user the one that created the issue in the first place, and for that you can easily create an obligatory custom field with the name of the reporter and other for his or her email.
You can also set JIRA to create issues via email and the reporter would get all the comments sent to his or her email.
I answered a similar question a couple of weeks ago, take a look:
If you're not familiar with setting permission schemes, here it goes:
Any further questions, just ask!
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