atlassian-jira-outgoing-mail.log -- once you have configured Outgoing Mail, you can enable the Outgoing Mail log and Debugging (JIRA Administration >> System >> Logging and profiling):
Also, you can add and increase the logging level for certain packages (e.g: com.atlassian.mail ) to get more details logged.
Finally found the inbound log after doing some digging. But it doesn't look like it has outbound emails on it. There is a feature request at https://jira.atlassian.com/browse/JSDCLOUD-4698.
Jira Settings > Products > Email Requests. On the email requests page, you'll find a list of email addresses. To the right (under "Actions" column) will be "View Log."
Atlassian, why don't you put this in the System > Mail section next to Global mail settings, Outgoing mail, Incoming mail and Send email?! >_<
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