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!
Hello Atlassian Community! I'm Teresa, the Product Marketing Manager for Jira Service Desk Server at Atlassian. Curious about the latest updates in the Jira Service Desk Server...
Connect with like-minded Atlassian users at free events near you!Find a group
Connect with like-minded Atlassian users at free events near you!
Unfortunately there are no AUG chapters near you at the moment.Start an AUG
You're one step closer to meeting fellow Atlassian users at your local meet up. Learn more about AUGs