Summary

Currently it's not possible to search for comments in a way that only public or internal comments will be filtered. In many cases this would be useful for project admins and/or service desk agents to manage the service desks better.

Suggestion

Provide a JQL function which is able to differentiate between public and internal comments.
 

Workaround

The suggested workaround is to make use of the database, we have a KB here for how you can do this: How to Retrieve Internal or External Comment in Jira Service Management