JIRA Authentication Problem

I've already posted this question as a comment at https://www.jetbrains.com/help/hub/1.0/Atlassian-Jira-Auth-Module.html?origin=old_help, so sorry for duplication - I hope that community can help...

JIRA authentication does not work in my case. When I try to "test login" I'm constantly getting "JIRA Server processing exception" message. Similarly, when I skip "test login" and simply save the authentication module, when a user tries to login he/she gets message "Authentication failed. JIRA Server processing exception." message.

I've enabled CORS by following these instructions, and it looks that integrating JIRA issue tracker with a particular projects went well. (On the project settings page, "test connection to JIRA" click reports "Connection to JIRA was successful".) It also creates project-associated JIRA auth module, but again this module does not work - it reports the same "JIRA Server processing exception" message.

I have freshly installed JIRA 7.1.4 on-premise, with also freshly installed Upsource on the same Ubuntu 14.04 machine.

If you need some logs for troubleshooting this please let me know exactly which log files you need, and I will attach.

Thanks!