Sync Bridge runs on Atlassian Forge. Connectors, sync rules, the links between work items, and sync history are kept in the app's storage on Atlassian's platform. Connector credentials are stored as encrypted secrets.
Each work item's sync status is also saved as a property on the work item, which powers the panel and JQL search. Work item data is sent only to the other Jira Cloud site you connect.
See also: Installation Guide (permissions the app requests)