The downside of anonymous speech should be fairly obvious to anyone who has ever read the comments section of pretty much anything on the internet. There is something about anonymity that brings out the worst in some people, and there were newspaper trolls long before there were Internet trolls.
The upside, on the other hand, is a little more difficult to see.
In fact, at the time of America’s founding, many of the original supporters of the Constitution argued against anonymous speech.
“As every American has a right to his own sentiments on the subject [the Constitution], so he must have liberty to publish them,” wrote one pro-Constitution essayist in the Boston Independent Chronicle on October 4th, 1787. “Yet he cannot be a friend to his country,” the writer continued, “who upon a production on the subject, will conceal his name.”
The debate over the Constitution, to the apparent chagrin of this writer, was done almost entirely through pseudonyms. Most writers, on both sides of the question, chose to conceal their names from the public, as was the custom in the late eighteenth century. A free press, one of the most essential principles for the founding generation, was presumed to include a right to anonymity.
The American founding generation had this crazy idea that ideas mattered more than who espoused them, but politics has a funny way of interfering with even long-standing traditions.

The American founding generation had this crazy idea that ideas mattered more than who espoused them.”]
The supporters of the proposed Constitution, who called themselves Federalists, wanted to expose their anonymous opponents, who they dubbed (rather unfairly) Anti-Federalists. In pro-Constitution areas, which included all of the major cities, to oppose ratification was dangerous to one’s political career. The shield of anonymity was a useful one, allowing for free exchange of ideas without fear of repercussions.
In many cases, when Federalists learned the identity of opposition writers, they would set out to destroy them economically and politically by spreading their identity far and wide. It was doxing, 18th-century-style, and it had the predictable chilling effect: essays against the Constitution became rarer while essays in support continued to show up. This was not a reflection of increasing public support for the document. In fact, there is no evidence that its support grew much during the course of the debate.

Partisan Hostility

Today, we still argue over just what freedom of the press entails. In an era of strong partisanship, in the wake of an election that broke up friendships and divided families, the idea of being able to speak one’s mind without revealing one’s identity is surely appealing.
We have a president who attacks even partisan allies who oppose or criticize him, tweets about “FAKE NEWS,” and dismisses “so-called judges” who stop his actions. We have members of that administration suggesting that the president’s use of power “will not be questioned.” We have efforts to shut down opposition wherever it arises. All of this can have a chilling effect on political speech.
This is not just a Donald Trump problem. The level of partisan hostility has been high for a while now, and it has become socially perilous to speak one’s mind in the wrong company. Democrats shout down Republicans just as Republicans shout down Democrats, and political discourse turns into angry recriminations and accusations rather than serious discussion over policy.
None of this is really new. One cannot read the debates over ratification without noting the similarities. Most Americans, if they are familiar with those debates at all, have read the Federalist essays and nothing more, but the Federalist arose out of a nasty debate full of personal attacks and partisan hyperbole. The Federalist, like the best of the opposition essays, represents the power of ideas to rise above the mire of partisan battle.
People with minority opinions frequently perceive danger in speaking their mind, and those who speak against the grain often face negative consequences for their courage. The possibility of anonymity can help those afraid to speak up to use their voices. The idea is not speech without consequences, but rather a fuller exchange of ideas where the marginalized and outnumbered can share their position without fear of recrimination.

The Dilemma of Free Speech

The argument for revealing names in 1787 was dubious at best. “A Citizen,” writing in the Massachusetts Gazette, suggested that publishing author names was “perfectly reasonable” and “perfectly consistent with the liberty of the press.” He then revealed his hand, suggesting that it was especially important for anyone opposed to the Constitution to be known.

Even though anonymous speech may entail a loss of civility, it expands freedom.”]
The problem with this Federalist argument was that the press is not really free if there are substantial personal repercussions for expressing an unpopular opinion. Truly anonymous speech offers a solution for this, even as it facilitates hateful and personal attacks.
This is the dilemma of free speech and a free press; we either need to facilitate expression, even of unpopular ideas, or we need to stifle ideas that we as a society deem harmful. We cannot have both. Even though anonymous speech may entail a loss of civility, it expands freedom.
The American Founders sided with expanding freedom. The first Congress did the same, in passing what became the First Amendment, and most of the states agreed by ratifying the amendment. Reading the public debates in the late 18th and early 19th centuries, and comparing them with our own, it is hard not to conclude that the quality of public discourse has declined sharply. Perhaps the time is ripe for a renewed emphasis on ideas over personalities, if not a return to pseudonymous public writing.