Just because the court rules in favor of the victorious party does not mean that it is the victorious party.
For the 1st amendment to be invoked, the federal/state/local government has to be on one side of the "vs." in the case name. Otherwise, it's a normal civil case between two private parties, and the appropriate laws apply.
Because the government has no way of forcing the third party to do its bidding. If the government were to pass a law saying "Newspapers should not print any op-eds critical of the government", for example, then it becomes a first amendment issue and the newspaper can sue for the right to print such things. (Which has happened, BTW.) But if the newspaper decides on its own not to print such op-eds, it's perfectly within its rights and there's no first amendment issue.
Similarly, the government can pass a law allowing copyright holders to protect their intellectual property - say, the DMCA. But it's then up to the copyright holders to bring lawsuits against people who publish their work - the government can't force them to do so, and they're completely within their rights to release their work under the GPL or Creative Commons. No free speech issue - it's a property issue.
IANAL, but listen to the law student above. His comments basically square with my understanding from lawyer friends and civics classes and occasional reading of court cases.
From a political philosophy perspective, you could look at it as libertarianism vs. anarchy. The libertarian POV is "You can do anything you want, as long as it hurts no-one." The anarchist POV is "You can do anything you want." The Bill of Rights is intended to protect libertarian ideals, not anarchist ones. It does not give you license to say anything you want, it prevents the government from arbitrarily restricting what you can say. The government can and should still make laws preventing your speech from hurting others.
For the 1st amendment to be invoked, the federal/state/local government has to be on one side of the "vs." in the case name. Otherwise, it's a normal civil case between two private parties, and the appropriate laws apply.