Aside from copyright (and abuses and highly dubious interpretations of it), some consideration in cases of this kind also needs to be given to privacy laws and rights in various jurisdictions. I don't know much about rights to privacy in the USA where the site is based, but within countries which are signatories of the European Convention on Human Rights, this case would seem to me (IANAL) to be covered by article 8 of that convention:
Article 8 – Right to respect for private and family life
1. Everyone has the right to respect for his private and family life, his home and his correspondence.
2. There shall be no interference by a public authority with the exercise of this right except such as is in accordance with the law and is necessary in a democratic society in the interests of national security, public safety or the economic well-being of the country, for the prevention of disorder or crime, for the protection of health or morals, or for the protection of the rights and freedoms of others.
Likewise, article 12 of the Universal Declaration of Human Rights reads:
No one shall be subjected to arbitrary interference with his privacy, family, home or correspondence, nor to attacks upon his honour and reputation. Everyone has the right to the protection of the law against such interference or attacks.
It seems fairly clear that E2's private messages constitute a form of "correspondence", (although I am sure that dissertations could be written on the subject if one felt like it). It has certainly been applied to email, including email to closed mailing lists (the existence of an archive seems to be a key distinction between private correspondence and published matter in such cases, which may be of significance as regards the catbox, but that's a whole nother topic for later.)
Although the European convention is not the strongest piece of international law out there, it has been transposed in various ways into the national legislation of many signatory countries - here in the UK by the Human Rights Act (2000). Many countries also have separate legislation designed to protect private data stored on computers by organisations, businesses and public authorities, which might be construed to cover the private messages held on E2's servers. Although the likelihood of E2 ever being caught up in legal proceedings under any of these laws is obviously pretty small, it seems to me that their existence gives a fairly clear indication of what ought to be considered a sane approach to such matters: that, in the absence of pressing reasons in the general interest, the disclosure of the private correspondence of a third party is unacceptable, quite regardless of the existence any copyright on the text.