I think it is understood that Mattermost doesn't natively integrate with JIRA – I believe that integration is accomplished by changing an open-source Slack integration. Are there specific ways that Atlassian tools can integrate with Mattermost? What would these be?
Typically, a software is created to integrate JIRA with any other software. A quick search in the Atlassian Marketplace (https://marketplace.atlassian.com/search?q=Mattermost) says nothing about a Mattermost related add-on. So your best bet here is to create one yourself.
It's not that hard to integrate with JIRA. You can create an external software or an internal one (install it in JIRA itself and install it in Mattermost). I will do the external one, a piece of program that does not need to be installed. All it does is send/receive information from JIRA and Mattermost (that is, if Mattermost have an External API to talk to).
I am thinking Mattermost should be able to consume the webhooks data posted by JIRA. You may just need to define a webhook in JIRA. But, my local tests currently does not work when I try to post it to mattermost at the moment. At the same time this blog says JIRA webhooks should work - http://www.mattermost.org/mattermost-v1-1-open-source-slack-compatible-integrations/
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