Is there any way to share cookies between my browsers, so whatever I do on one browser also takes effect on the other one? (Chromium and Firefox)
No, you cannot share cookies across web browsers. At present, there are no services that synchronizes cookies just like how bookmarks are synchronized.
An alternative approach you may want to consider is to use an extension such as Swap My Cookies to manage multiple sessions within a single browser.
Xmarks (http://www.xmarks.com/) is an extensions so that you can share bookmark, history, password across browser or computer. It works in Chrome, Firefox, IE in both Window and Mac.
Is is the extension you're finding?