Trouble with session ID and Drupal cookies when trying to establish a connection with Service Module and http-post
All of the cookie information is set and returned in the response header of the login call.

This might also be of some help which explains how to make authenticated calls via a remote client.
http://drupal.org/node/910598

Also note that if you use a Firefox plugin, it will be able to access the cookie set in Firefox. Thus if you use a different application that cookie will be different, so you will not be able to auth via your custom app and simultaneously see yourself as logged in in Firefox.