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.