you could consider an addon like Automation for Jira.
The users haven't a account in your jira ?
If a users don't have the browse permission in your permission scheme you cannot send him a notification.
If he can see the issue or you have the browse permission set to Anyone, then you just have to work with you permission scheme that allow you to send an email to a specific project role, single adresse mail, groups and so on.
Hope this helps.
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