I just checked with my testing account which does not any moderator permissions, and I could edit the text in the conversation (PM) via the edit button below it. Does this not appear for you? If you are talking about posts on your profile's wall, then those cannot be edited, except by a moderator with the explicit permissions to do so.
From what I could check in my own conversations, you can only edit the title. So pretty much any edit to the conversation itself is in form of another message. Dunno if the edit button is hidden somewhere or what lol.