I think this would be an interesting idea. As the owners and creators of the site live in California time then local Chess.com time would be Pacific Standard Time (PST).
Does anybody know what time it is?

Central site time can be useful on an auction site, to help participants synchronise with the end of an auction. I would have thought that showing the two individual local times of two players to each other would be more relevant on chess.com. And if you need to know that, just ask Mr. Wolf. http://www.chess.com/forum/view/community/whats-the-time-mr-wolf
Would it be possible to place a gadget that gives the current, official, Chess.com time and date the home or entry page?