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:
...be more productive while being fun to use at the same time. For some, getting started can be a bit intimidating. This is especially true if Jira Service Desk is your first exposure to Atlassian...
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