Jira Server / Data Center
Simple answer: no.
Everything is handled locally between Outlook and Jira and required information are stored on Jira and in the issues. See Data Stored in Jira for more detailed information which data are stored.
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.
Jira Cloud
In Jira Cloud we do store administrative settings on our servers. This includes administrative settings and information about your Jira Cloud instance.
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 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.