This is absolutely not true under federal law (18 U.S. Code § 2511.) (also apparently CA PC 631).
It doesn't matter who owns the facilities. Except for certain narrow exceptions (warrants, government officers, jails, incidental operations necessary for maintenance) any unauthorized third party interception of a private communication is unlawful. Common carrier status is relevant for liability but is irrelevant for § 2511.
I think you are confusing a Twitter DM with a private communication. Note that Twitter is not intercepting your DMs, they are the ones transmitting and storing them.
It doesn't matter who owns the facilities. Except for certain narrow exceptions (warrants, government officers, jails, incidental operations necessary for maintenance) any unauthorized third party interception of a private communication is unlawful. Common carrier status is relevant for liability but is irrelevant for § 2511.