Not quite what you're describing, but the venerable Squid cache can do distributed caching. Something, at least, to play with with your nerdy friends.
(Also, the mere fact such a scheme would have to jinx with the mechanics of a fundamental browser security mechanism should be enough to indicate how difficult it would be to implement safely!)
Consider that SSL is largely used for connection encryption, though it has the additional side-effect of site authentication.
If you can still validate the site, and rely on a hash of the content to detect changes, then you're starting to get toward a cacheable, secure, autenticated, system.
I need to think more about this.
That said, the web site[1] offers great docs with lots of config samples to start from. If you're starting from scratch, take a look at CARP configuration. Squid also speaks ICP and HTCP.
Squid is one of the rarely-sung heroes of web content delivery.