It kind of depends if they are using WID/SQL and are loadbalanced etc.
If you are using WID the configuration between the nodes is taking place via pull-method.
The secondary-servers are reaching out to the primary member and retrive the latest copy of the primarys WID database/configuration. The sync between the nodes goes over port 80.
There is nothing "cached" on the servers during authentication (more then normal login events etc).
So the login information/event will only show up on the server that actually handeled the login request from the user.
You do not need to run any clean up-stuff, all the vital information for making auth/SSO works is stored in a cookie in the users browser on the client.
