I have a service desk project created and added few users with admin and service desk customer privileges. Some of the users who have those privileges cannot change anything on the tickets though they can view them. Can't figure out what admin privilege they have in this case.
If they do not have a Service Desk Agent license and are not part of the service desk team role in the project then they will not be able to edit the issue.
With Jira Service desk only agents are allowed to edit tickets, service desk customers can view the tickets but are not able to make edits to the tickets post creation.
This previous question and answer has some links to help provide more detail:
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