Personally I find the UI difference between PMs and public posts on the website to be far too small. They are basically identical. I suppose that’s a different design issue.
But in my email interface, once I am reading there is nothing else telling me that the message is from a PM and not a public post. And if someone contacts me using the message button in a public thread, then the email subject line will by default be identical to the public thread title, except for the addition of “Re:” which in email land doesn’t mean anything other than that it is a reply.
But I guess your question is really about if it matters if I know if someone is sending me something that is just between me and them, or if I am reading something that is between me, them, and the whole internet. Personally I’m not in the habit of replying by email, but If I was, I would want the distinction to be very, very clear.