Going by the description, if you know a file's plaintext, you know both its plaintext hash (decryption key) and its encrypted hash (storage key). The communication to request files uses the storage key. It would be rather easy to identify by statistical means, people who request files of a common type, for example known CP images. Forwarding a few requests for CP or having it stored doesn't indicate anything. Reading or writing a lot of known CP would be probable cause for a search. This is especially easy because the majority of CP is not new and FBI presumably has the whole damn lot on a hard disk somewhere. They can easily pre-compute hashes and just sit monitoring.
As a result, this network would be useless for CP-mongers, they'd get caught about as easily as using plain old FTP.
On the other hand, I do think the ability to pre-compute hashes is a flaw that massively reduces this network's usefulness to dissidents. It is quite effective as a "publish a manifesto network" with a secret writer and overt readers - the original write is of a never-before-seen file, there is no hash to monitor. It's unsafe as a "store my pirated stuff" network, writers of a well known file can be tracked. And it would take very little statistical monitoring to reveal the interests of a reader.