If an individual replies to an email it should create a comment. There are cases where a user would not be able to reply to an email and this to work. If you have the service desk setup so customers have to be manually added to the service desk then a customer could be cc'd on a request and when they reply it will not create a comment.
Can you share how your service desk is setup?
You might also walk through this to ensure everything is setup properly. https://support.atlassian.com/jira-service-management-cloud/docs/receive-requests-from-an-email-address/
The assignee should also create comments on the issue when they send an email to the service desk. As long as they are licensed as an agent and have permissions to the project they will be good to go. The fact that they are the assignee should almost guarantee that.
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