This could be the ultimate in sensible default behavior, and could work for anons as well as logged in folks.
Issues to consider:
document.write inline vs Dom manipulation -- compat, purity, display/flicker issues?
Formatting and localization -- we have a lot of date and time formatting options, including alternate calendars. Do we need to replicate all of these in JS code or can we get away with just the default -- assuming any other option would force server-side rendering?
Date headers in lists such as RecentChanges will move depending on the timezone.
Can we handle talk page sigs? (or leave that to LQT?)
How should we mark timezone on output to indicate to the user what he/she is looking at?