It has recently been something of a problem on the intarwebs that when people's stream of offensive, tacky, cliche-ridden drivel is challenged, they resort to "FREEDOM OF SPEECH" as a defense. There is some confusion about what exactly constitutes Freedom of Speech in some quarters, and one of the subtropes of this discussion is the idea that the first amendment to the constitution protects anonymity in speech. Some months ago, an administrator on the popular site Reddit, who had posted a variety of things that were offensive in a variety of ways, was "outed", with his real identity revealed to the internet. This caused some negative repercussions in his life, and some people felt he had a legal right to anonymity.
The issue is fairly easy to explain, and the confusion arises because people confuse two different things. People can not be legally prevented from publishing what they want anonymously or pseudonymously. But they have no legal protection from private people from researching or publicizing their identity.
In 1995, in the case of McIntyre v. Ohio Elections Commission, the Supreme Court decided, in a 7-2 decision, that an Ohio regulation that required that any electoral materials disclose the identity of their author was not constitutionally valid. While the court agreed that Ohio had a reason to possible censor electoral materials against false or libelous statements, they further decided that a law prohibiting anonymous campaign literature did not further this cause and was a violation of the First Amendment right to free speech. The entire reasoning can be found around page 410 of this large pdf. So the Supreme Court has decided that the government can not prohibit anonymous speech.
On the other hand, this doesn't mean that other people are prohibited from researching or publicizing the identities of those who try to publish anonymously. These are two different issues, and the conflation of them has caused some confusion, which hopefully I have cleared up here. Government entities can't prevent you from speaking anonymously, but private citizens have no legal obligation to protect your attempts at anonymity.