Surrendering a copyright is like surrendering a child to adoption: You surrender all rights of custody and control over your work.
I declined to sign in exchange for a paltry payment and the presumed thrill of seeing my byline in the New York Times and stopped writing for that august publication, but I suspect I was in a tiny minority. The Times was hardly alone in treating freelancers like serfs. Publications routinely and successfully demand that freelancers give up all rights in their work in order to be published; online publications generally expect you to give up your intellectual property rights for free, instead of for the usual pittance. (The Atlantic, which, in my experience treats writers with respect, is an exception, which is why my posts appear here.)
Why not give in and give up all rights in your work? Surrendering a copyright is like surrendering a child to adoption: You surrender all rights of custody and control over your work. In fact, it's no longer your work; it's the property of the publication. You may not recycle, reprint it, or quote extensively from it in violation of fair use without the publication's permission. The publication's employees (editors) may rewrite what was once your work without your permission, altering your perspective as well as your language, including or omitting your byline at their discretion. Usually they disavow any intent to substantively alter your work, but I am always wary of people who make non-negotiable demands for rights they claim to have no intention of exercising. (When confronted with these contracts, I almost always offer publishers a perpetual license to reprint my work for free, but they almost always demand all rights, including the unilateral right to alter it, as well.)
As a general rule, federal law locates intellectual property rights in the creators of intellectual property, which essentially means that absent an express surrender of copyright, publishers have no rights to a freelancer's work, other than the right to its initial publication. What would New York Times editorial writers say about a corporation that required its consultants to cede all their rights under federal law as a condition of employment? I once posed that question to the Times, naturally to no avail. Media companies guard the sanctity of their own copyrights but tend to treat other people's copyrights with a mixture of acquisitiveness and contempt. When editors at the Times publish an op-ed stressing the cultural value of copyright protections, it's probably not your copyrights they have in mind.