It's like YouTube. Google refuses to take ownership of any content provided by their site -- even though in that case they do host it -- as they are merely a "provider". And even then, they must enforce a DMCA mechanism or face the consequences. Otherwise they would have been sued into obscurity.
>If Twitter does not comply with this request, and the Stolen Information continues to be disseminated by Twitter in any manner, SPE will have no choice but to hold Twitter responsible for any damage or loss arising from such use or dissemination by Twitter, including any damages or loss to SPE or others, and including, but not limited to, any loss of value of intellectual property and trade secrets resulting from Twitter’s actions.
Page 3 of Sony PDF
Sony should be sending these letters to their network department as they are the ones that appear to have 'disseminated' the Stolen Information.