So you need to look at your notification scheme in JIRA as perhaps you have it set up to do all sorts of notifications that are duplicating what the customer notifications are doing.
If that's not the case, perhaps you can be more specific on what action causes the notifications. Is there a customer notification and an internal notification when a comment is added? Are you the agent,reporter or being shared? What are the permissions set up like for the project?
I think Susan is on the right track here. With Service Desk it's important to understand there are two different possible notification types. These differences are explained in the KB JIRA Service Desk Notification explanation
The footer of the messages will tell you if this is Service Desk or Jira Core notifications.
I'd be interested to look closely at these "duplicates". I suspect that you're getting one message for a new comment added, and another notification because either the issue transitioned to a new state, or the issue was edited. These are different events that might happen at the same time. As such the notification schemes might be working correctly depending on the details of your situation. In these cases, theses are not true duplicate messages.
...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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