For example with box-shadow or transition, do I still need to use prefixes like "moz" or "webkit?" It seems... odd, and at least on Firefox on Unix, it works perfectly without any prefixes.
Short answer : Use Them because HTML5 and CSS3 are still under development and are not fully supported by any browser.
Other stuff you may like to read : As a developer, we need to see from user point of view, cross-browser compatibility is essential to maintain your website layout almost same in all browsers, more over it also gives professional edge to your work if your layout is consistent in all browser, using proprietary properties won't increase your page load time, and most of the users don't update their browsers to latest version, and so they get a shabby rendered page, so to be safe use them unless and until CSS3 and HTML5 are completely supported by all browsers.