Private institutions should be bound by the first amendment once they've reached a critical mass. The reason I say this is that, if a small number of entities control the flow of information, and they get away with picking and choosing who has the opportunity to be heard, then the first amendment is effectively made meaningless. I want to be able to choose what comments to allow on my block and, at the same time, I want to be able to express myself on a prominent platform in a way that may or may not offend the powers that be.
Big friendly looking corporations like Google shouldn't be used as proxies for the government to bypass the rights of Americans(or anyone for that matter). For whatever reason, people see corporations as being different from governments. What is a government but a corporation that owns land? What is a corporation but a form of government? They're quick to should about the importance of dissent, telling the government what it cannot do to our rights, yet corporations get a free pass when they are on the right side and hand out free goodies.
Beyond the likeness of corporations to governments, big corporate censorship should be opposed on the grounds of principle. America and other countries are founded on the principle of freedom of speech, among other things such as the concept that all "men" are created equal. We don't allow businesses to discriminate people by their race, color, sex, or religion because of that principle. Why then can we not apply the principle of freedom of speech to Silicon Valley and Big Media?
It's pretty frightening that the same people who told me how brilliant Nineteen Eighty-Four is(I found it the least convincing of dystopian warnings I've read), easily fall into the trap of believing that we desperately need a centralized Ministry of Truth. In the last year I've been realizing that Nineteen Eighty-Four is more realistic than I had believed.