In the current versions of Jira Service Desk Server, this is not possible to do with the native Jira code alone. There is an existing feature request to add in the event of an issue assignee changing over in https://jira.atlassian.com/browse/JSDSERVER-1222
I would recommend watching this issue for updates.
I did notice in the comments of this feature request that it appears there might exist plugins for Jira that can turn notify JSD customers of events like this. One of this appears to be the Jirassimo Emails Reminders Notifications. I have not used this plugin personally. Also it is possible there might be other alternative plugins in our Marketplace site that might also be able to offer similar functionality.
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