This concept may sound clever at first but gives you as the user no additional confidence compared to encrypting data on the server side upon arrival. Either way, you're trusting the host.
The threat model for server-side encryption is essentially:
1) the host has an unethical employee who wants to read your content.
2) the host's servers are insecure and get compromised.
3) someone successfully MITMs your connection to the host (possibly due to the SSL problems being discussed here).
4) the government compels the host to provide your data (i.e. what happened with Lavabit).
The threat model for browser-based client-side encryption is the same! In any of these cases, the attacker (or the host, in case of #1 or #4) simply sends JavaScript encryption code to your browser with a backdoor in it.
Cryptocat originally worked the same way: all chats were encrypted on the client side, but with JS code sent from the server, in which a backdoor could be inserted at any time. After much criticism, this is why Cryptocat is now a browser add-on, with discrete releases made available from a central source (Chrome Store/Mozilla addons site), which can be audited.