Update June 2019
Google is removing the ability to detect Private Browsing Mode permanently in Chrome 76 onwards. So, if you’re wanting to detect private browsing it’s now impossible (unless you find a way to do it that Google hasn’t found). The ability to detect private browsing mode has been acknowledged as a bug and was never intended.
To anyone else coming across this question, please note as of 2014, there is no reliable or accurate way to detect if someone is browsing in an incognito/private/safe browsing mode through Javascript or CSS. Previous solutions that once worked like the CSS history hack have since been rendered unusable by all browser vendors.
There should never be a situation where needing to detect private browsing mode on a normal day-to-day website is ever needed. People are choosing to browse anonymously and or not anonymously for their own reasons.
Browsers like Chrome and Firefox do not disable functionality like localStorage any more. They simply namespace it in a temporary location to prevent websites that use it from erroring out. Once you’re finished browsing, the namespace is erased and nothing is saved. If you are testing for localStorage support regardless of mode, it will always return true for browsers that support it.
Other means of detecting private mode in Chrome specifically have been completely patched and will no longer work.
If it is required internally by a company, you should develop a browser plugin. Chrome and Firefox, in particular, expose internal API’s which allow plugins to check if the user is in private browsing/incognito mode and action accordingly. It cannot be done outside of a plugin.