We had the same requirement and we use Jira 7.2.8. Only the username can be added in the "Alert User" section. To alert the current assignee of the issue you can add an internal comment which will trigger an notification email to the assignee of the request.
Hope this helps.
I know this is a old post but for others looking for a solution I thought I would share my solution.
There was a demand for our assignee's to get notified when a status had breached or was about to breach each status within our workflow. So I setup the following.
1. Setup your "When" rule based on your SLA.
2. Create the if for each assignee within your project.
3. Then create the Alert for each assignee.
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