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I share your concern, although I think it is more a misguided and foolish method of repayment for campaign contributions by the RIAA than an attempt at stiffling freedom of speech. The big problem in the short term is that no one is going to know where the boundries lie. There was an article in ars technica today about one of the sponsors in Congress having copyrighted news clips posted on his website and available for download. This would be forbidden, but there is an exemption for fair use, and anti-streaming provisions only apply when done willfully. So that says that if you didn't "mean" to infringe on a copyright, you didn't violate the law.
The problem is that owners/hosts of website are expected to police what is there, and I am darn certain that if they get a complaint from BMI/Sony/WB/whoever, they are not going to take the time to ask each individual if they really meant to violate the [proposed] law. They are likely just to shut down your site. Then you can try to pick up the pieces. And do you think it will be easy to get PayPal to reinstate your account if they get a complaint? I don't, judging for how hard it was to get a clear error resolved that PayPal made on one of my personal accounts.
My personal opinion about SOPA is that it is a terrible potential law. But generally people don't pay attention to the terrible things that the people they vote for do everyday in congress, unless it is one of the hotbutton partisan cultural issues that get people all fired up. Hopefully enough people will call and write their senators and representatives on SOPA (& the companion bill in the Senate). The one upside is that there is opposition from groups on the left and right side of the political spectrum, so maybe there is some hope.