The phrase “corporations are people” is somewhat ambiguous. It does not mean that every corporation is a person, it means that corporations are _composed of_ people. (Think “soylent green is people”, if it helps you understand.) The people in a corporation have rights, and thus any action taken by a corporation may be an expression of those rights. A corporate press release has just as much protection from the first amendment as any other statement a person might make.
The people in a corporation can and do get sent to jail, even if the corporation itself does not.