Blockchains can do this. But they create relatively absurd systems of who can/can't send messages based on how much they've mined. Block chains are HORRIBLE for encrypted messaging. Since your message is PERMANENTLY part of the chain. So if a encryption is broken, or your password gets leaked. ANYONE can read your messages.
In most cases a DHT is far simpler. But naturally some nodes can be evil and log metadata. This is just a risk, but it also exists in block chains. Furthermore DHT's are lower latency then block-chains.
Blockchains aren't a magic fix all solution for network consistency/privacy models. Outside of currency transactions it is a horrible distributed system model.
Also if you look at it in terms of the CAP Theorem:
Consistency (every read receives the most recent write or an error): Not True for block chains. All reads are dirty. The further back in time you go the higher the probability the read isn't dirty.
Availability (every request receives a response, without guarantee that it contains the most recent version of the information): 100% true.
Partition tolerance (the system continues to operate despite arbitrary partitioning due to network failures): Yes the system will continue to function. But you can suffer data loss when the partition is healed.
Sorta, but not. The only similarity would be that you download global state. But you could continously throw that state away if you don't need archiving of messages / long history.
