Piece of paper isn't common carrier. If I write a note to my wife and give it myself it is private because I wanted and expected it private. If I taped it to a billboard it means I no longer expect it private and it is public.
Despite phone companies being common carriers (and thus being required to carry communications without censoring them), the Supreme Court has made it clear that people should have no reasonable expectation of privacy when they relay communications through a pay phone. The reasoning was that you give up your right to privacy when you involve a third party to your conversation. See Katz v. United States, 389 U.S. 347 (1967).
So the two can coexist.
Only common carriers are forced to carry speech unimpeded, regardless of its content. Expectation of privacy is irrelevant to common carrier status. The former is about whether it can be intercepted. The latter is about whether it can be impeded.
More recent decisions suggest that the third party doctrine, according to which there is no legitimate expectation of privacy in information voluntarily turned over to third parties, is unlikely to apply to private messages on Facebook:
As you said, Katz did say that we do have a reasonable expectation of privacy with respect to person-to-person communications. However, that has no bearing on whether the carrier can or cannot block the communication. That still hinges on whether the carrier is a common carrier.
It is according to Facebook. Here is a quote from their own website:
> We’re dedicated to making sure Messenger is a safe, private, and secure place for you to connect with the people who matter.
