Pretty-simplified answer, is that Zeitgeist logs your files accessed, sites visited, etc. (your computer-use-actions) and stores them LOCALLY. If your password is vulnerable, then the Zeitgeist database is just as vulnerable. Several Linux programmers have deeply investigated 'is Z uploading anything on you', and the answer is no, it is not. It is to facilitate searches and 'best-fit' to your local query only. That said, if you are using Firefox, Chrome, etc. as a browser, your browser is already logging your actions and THEY ARE being uploaded, depending on the site you visit and what they want to know relative to metadata.
If you want to turn your local Zeitgeist off, simply go to SYSTEM SETTINGS>SECURITY AND PRIVACY SETTINGS (In 14.04LTS, your version may vary), and then on the Files and Applications tab, turn off 'Record Files and Applications' button. If you're really concerned, delete your history there, too. This clears the database. Now, Zeitgeist is resident, but turned off.
My paranoia stems far more from the logging of ANY browser I use, then from Linux or Ubuntu.