Here is Rep. Donna Edwards (D-Md.) explaining why she is introducing a constitutional amendment in response to Citizens United:

Justice Brandeis got it right: democracy, or wealth concentrated in the hands of a few, but not both. And that's truer today as the United States Supreme Court just wiped away decades of legal precedent, allowing corporations to spend unlimited money from their treasuries on our elections. The American people already believe that corporate special interests and their lobbyists run the show around here. I mean, the halls are crawling with them. But that's not enough. Now the Court says to the big banks, to the drug companies, to the insurance companies, "Hey, all bets are off, and it's open season. Our elections are for sale."

So that's right, if the [corporations don't] like what this congresswoman is doing, they'll just forget the voters, buy TV ads, send robocalls, send a lot of mail, and beat her in November. A law won't fix this; we have to fix it in the Constitution. So today I'll introduce a constitutional amendment so that we the people can take back our elections and our democracy. This is not the People's House Inc. We are the people. It’s our house, it's our Constitution, and it's our election. And we plan to take it back from the United States Supreme Court.

Shorter version: We need to amend the constitution because the public already has a low opinion of Congress, and allowing people to criticize us more will make us look even worse; I might even lose an election. Not the most compelling reasons, assuming you are not a member of Congress. Here is the text of the Speech for People Amendment:

Amendment XXVIII

Section 1.  The sovereign right of the people to govern being essential to a free democracy, the First Amendment shall not be construed to limit the authority of Congress and the States to define, regulate, and restrict the spending and other activity of any corporation, limited liability entity, or other corporate entity created by state or federal law or the law of another nation.

Section 2.  Nothing contained in this Article shall be construed to abridge the freedom of the press.

Section 2 is presumably intended to protect media corporations (such as those that own the corporation-bashing New York Times and Washington Post) from whatever restrictions on "spending and other activity" Congress dreams up. According to Free Speech for People, which put together the amendment and provided the video of Edwards, the amendment "will do nothing to infringe freedom of speech or of the press." How so? "The First Amendment clearly prevents government suppression of 'the press,' whether a corporation or not, and that is as it should be." The First Amendment also says "freedom of speech," with no qualifier indicating that it does not apply to individuals organized as corporations, but they want to change that part. Why not restrict "freedom of the press" too?

In any event, the distinction means that officially recognized members of the media get to speak more freely than mere citizens who join together as corporations, even (especially?) if the main goal of their group is to affect public policy. If you work for The New York Times or The Washington Post (or Reason—whew!), you can speak your mind without worrying about how soon the election is or whether you are making voters more or less likely to support a particular candidate. But if you work for the National Rifle Association, the American Civil Liberties Union, the Sierra Club, or a small business, you'd better watch what you say when you speak on behalf of your employer. Aside from the blatant injustice of this two-class approach to the First Amendment, there is the problem of telling the difference between a media corporation and a nonmedia corporation, as the Supreme Court noted in Citizens United:

There is no precedent supporting laws that attempt to distinguish between corporations which are deemed to be exempt as media corporations and those which are not. "We have consistently rejected the proposition that the institutional press has any constitutional privilege beyond that of other speakers."...With the advent of the Internet and the decline of print and broadcast media, moreover, the line between the media and others who wish to comment on political and social issues becomes far more blurred.

The law's exception for media corporations is, on its own terms, all but an admission of the invalidity of the antidistortion rationale. And the exemption results in a further, separate reason for finding this law invalid: Again by its own terms, the law exempts some corporations but covers others, even though both have the need or the motive to communicate their views. The exemption applies to media corporations owned or controlled by corporations that have diverse and substantial investments and participate in endeavors other than news. So even assuming the most doubtful proposition that a news organization has a right to speak when others do not, the exemption would allow a conglomerate that owns both a media business and an unrelated business to influence or control the media in order to advance its overall business interest. At the same time, some other corporation, with an identical business interest but no media outlet in its ownership structure, would be forbidden to speak or inform the public about the same issue. This differential treatment cannot be squared with the First Amendment.

Obviously, if we incorporated the differential treatment into the First Amendment by changing the Constitution, then it could be squared. But that would not make it fair or wise.

Today Stanford law professor Lawrence Lessig and his group, Change Congress, endorsed the idea of amending the Constitution to overturn Citizens United, although it's not clear what language he supports. Previous Reason coverage of the idea here and here.