Not all speech is protected speech. Graffiti is speech, and the words being spoken could be argued as protected, but the act of spraying other people properties with it is not protected. Free speech rights does not overwrite other rights. As a defense in a court I would not bet my money on free speech in order to get away with crimes that happens to involves speech.
Historically the US court has defined speech into multiple different categories. One of those are called fraudulent speech which is not protected by free speech rights. An other category is illustrated with the anti-spam law in Washington State, which was found to not be in violation of First Amendment rights because it prevent misleading emails. Washington’s statue regulate deceptive commercial speech and thus passed the constitutional test. An other court ruling, this one in Maryland, confirmed that commercial speech was less protected than other forms of speech and that commercial speech had no protection when it was demonstrably false.
In theory a spammer could make non-commercial, non-misleading, non-fraudulent speech, and a site like twitter would then actually have to think about questions like first-amendment. I can't say I have ever received or seen spam like that.