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Everything is handled locally between Outlook and Jira and required information are stored on Jira and in the issues. See Data Stored in Jira https://yasoon.atlassian.net/wiki/spaces/M365fJ/pages/2330985432/Emails%2BData%2Bstored%2Bin%2BJira for more detailed information which data are stored.
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That being said, all issue- and user-related information are handled locally and are not stored outside Jira. You can check Data Stored in Jira https://yasoon.atlassian.net/wiki/spaces/M365fJ/pages/2330985432/Emails%2BData%2Bstored%2Bin%2BJira for more details.
The Outlook add-in stores all required data in the users-mailbox locally in Outlook. This makes sure the user is automatically logged in on all devices.