Is Gossamer Up?
@megtastique points out that two factors doom the whole design:
There's no way to remove content from Gossamer once it's published, and
Gossamer can anonymously share images.
Combined, these make Gossamer the perfect vehicle for revenge porn and other gendered, sexually-loaded network abuse.
This alone is enough to doom the design, as written: even restricting the size of messages to the single kilobyte range still makes it trivial to irrevocably disseminate links to similar content.
Protected Feeds? Who Needs Those?
Gossamer's design does not carry forward an important Twitter feature: the protected feed. In brief, protected feeds allow people to be choosy about who reads their status updates, without necessarily having to pick and choose who gets to read them on a message by message basis.
This is an important privacy control for people who wish to engage with people they know without necessarily disclosing their whereabouts and activities to the world at large. In particular, it's important to vulnerable people because it allows them to create their own safe spaces.
Protected feeds are not mere technology, either. Protected feeds carry with them social expectations: Twitter clients often either refuse to copy text from a protected feed, or present a warning when the user tries to copy text, which acts as a very cheap and, apparently, quite effective brake on the casual re-sharing that Twitter encourages for public feeds.
DDOS As A Service
Gossamer's network protocol converges towards a total graph, where every node knows how to connect to every other node, and new information (new posts) rapidly push out to every single node.
If you've ever been privy to the Twitter “firehose” feed, you'll understand why this is a drastic mistake. Even a moderately successful social network sees on the order of millions of messages a day. Delivering all of this directly to every node all of the time would rapidly drown users in bandwidth charges and render their internet connections completely unusable.
Gossamer's design also has no concept of “quiet” periods: every fifteen to thirty seconds, rain or shine, every node is supposed to wake up and exchange data with some other node, regardless of how long it's been since either node in the exchange has seen new data. This very effectively ensures that Gossamer will continue to flood nodes with traffic at all times; the only way to halt the flood is to shut off the Gossamer client.
Passive Nodes Matter
It's impractical to run an inbound data service on a mobile device. Mobile devices are, by and large, not addressable or reachable by the internet at large.
Mobile devices also provide a huge proportion of Twitter's content: the ability to rapidly post photos, location tags, and short text while away from desks, laptops, and formal internet connections is a huge boon for ad-hoc social organization. You can invite someone to the pub from your phone, from in front of the pub.
(This interacts ... poorly with the DDOS point, above.)
When a user enters a new status update or sends a new private message, their Gossamer node immediately forwards it to at least one other node to inject it into the network. This makes unencrypted Gossamer relatively vulnerable to traffic analysis for correlating Gossamer identities with human beings.
Someone at a network “pinch point” -- an ISP, or a coffee shop wifi router -- can monitor Gossamer traffic entering and exiting nodes on their network and easily identify which nodes originated which messages, and thus which nodes have access to which identities. This seriously compromises the effectiveness of Gossamer's decentralized, self-certifying identities.