Sorry to say that you can't customize the Service Desk notifications as you can in JIRA. For this we have the following suggestion where I'd advice you to set yourself as a watcher to receive new updates about this.
In additional and in case if you want to create different notifications, please check the Automation rules out as the following documentation:
...word of mouth, so by 2016, we were working with several other entities on campus to implement Jira Service Desk . The Atlassian motto of “for every team” has really come true for us in this case. We...
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