Interesting question.
I'll answer with a few more questions:
- Would it be relevant what the fine print says on Party B's website?
I do not think so, because you might send something to an employee of Party B using his Party B email address, without ever visiting their website.
- Would there be any difference if Party A sent the email marked as personal or confidential?
I guess that would mean that there is some breach at least of common courtesy.
- If a journalist would find a letter on her desk (or anywhere else for that matter), why would she not publish it if she wants to?
It is probably up to the creator of private communication (love letters or confidential documents) to keep it private, and when somebody gets possession of it by legal means, then the "finders keepers" rule might come in.
So I think, to answer your question, if the document was never requested to be private, then I do not think it is. Having said that, depending on the situation, common courtesy would require that Party B would remove the email on a reasonable request by Party A.
And that is forgetting about any copyright issues, if the email would contain an original article for example. That would probably be clearer, you would be breaching copyright if you published without consent.
Cheers.