I can think of so many reasons why this won't work... Where to start:
Messaging clients nowadays use centralized servers, and users only download data that's relevant for them. Downloading all posts from a forum will easily fill tens of gigabytes, and it grows continuously.
Satoshi isn't around, so giving Satoshi access to any data won't work.
Gmail is a bad example: any POP3 email client used to have all emails available offline. But they all relied on a centralized server. By now, being online isn't a problem, but sharing your IP-address with all other forum users (just like when you use Torrents) is not a good practice.
A forum that doesn't allow administration and censorship is the recipe for disaster. Although the forum, as it is now, allows freedom of speech, it still prohibits spam, plagiarism and spreading malware. Probably more too. And hundreds of posts (at least) are removed daily because of this.
On the other hand, a forum like you propose will be the ultimate garbage collector.
Theymos has posted some good ideas about a decentralized forum. I remember the idea of ignoring users based on a system similar to the current Trust list. Without central moderation, everyone will have to choose for themselves who's ignore list to follow.
As much as I like the idea, there's a reason I haven't seen any real decentralized forum. The cons don't outweigh the pros.