As a follow-up to 169367, I think that user should receive a notification when his e-mail is converted into issue.
Can it be achieved somehow without turning on notifications about "My Changes"?
I think the best solution for you would be to use the JIRA Enterprise Message Handler plugin, (JEMH), which would allow you to tailor notifications and also send notifications to non-JIRA users who create issues in the application via email, so I'd recommend you give it a go and see whether it fits your needs or not: https://marketplace.atlassian.com/plugins/com.javahollic.jira.jemh-ui
All the best,
JEMH has two approaches to notification of issue creation, your description seems to match the simpler JEMH Configuration approach, for non-JIRA users, this means they can get a 'thanks, we got it' message, and hear nothing more. You can optionally use both approaches, enabling say the ISSUE_RESOLVED event to let them know you're done.
There is the ISSUE_CREATED event which fires when issues are created, but your JIRA users may have opted out of telling them what they just did (gets old quickly). If you want to add a postfunction to the ISSUE_CREATED event you could probably script something, see script runner.
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