Bee communicates solely with your designated server.
No intermediary servers are used. Bee smartly caches, polls and fetches deltas whenever possible from your server's API.
Your Mac is the only computer connecting to your server.
See here for a high-level overview on how this works.
When you enter your authentication credentials in Bee, they are securely stored in your Mac's Keychain locally.
Each service that Bee supports has different requirements for authentication due to differing APIs.
Bee stores your username and password securely in your Mac's Keychain. Bee uses the JIRA REST API to maintain a cookie-based session. This provides ease of use when setting up Bee to work with your server, and also maintains a reliable, always logged-on connection to your JIRA server. Your session is automatically managed by Bee and renewed whenever needed, transparently.
Upon first authenticating with FogBugz, Bee retrieves a unique token and then stores this securely in your Mac's Keychain. Your password is not stored.
Bee retrieves an OAuth token from GitHub when it first authenticates with your server and then stores this securely in your Mac's Keychain. Your password is not stored.