The issue of Internet misogyny has received a great deal of attention in recent weeks, between the feminist website Jezebel battling rape GIFs repeatedly posted to their comments section and videogame critic Anita Sarkeesian having to leave her home after a series of Twitter threats that included her home address. There is a common assumption that the targets of such vile behavior are overwhelmingly women who are abused because they are women—to the point where “women aren’t welcome on the Internet,” as Amanda Hess argued in a widely discussed article in Pacific Standard magazine this year. Reviewing women’s online tribulations in the last month in The Daily Beast, Samantha Allen asks, “Will the Internet ever be safe for women?”

At the same time, there was little reaction to a report contradicting the narrative that male public figures get considerably less Twitter abuse than their female counterparts. While the study, conducted by the British think tank Demos, was limited to a fairly small sample of British celebrities, journalists and politicians whose Twitter timelines were tracked over a two-week period, its findings are nonetheless interesting. On the whole, 2.5 percent of the tweets sent to the men but fewer than 1 percent of those sent to women were classified as abusive. Male politicians fared especially badly, receiving more than six times as much abuse as female politicians.

The only category in which women got more Twitter abuse than men was journalism: abusive messages accounted for more than 5 percent of the tweets sent to the female journalists and TV presenters in the study and fewer than 2 percent of the ones sent to the male journalists. (However, the most abused male journalist in the sample, controversial ex-CNN host Piers Morgan, was counted as a “celebrity” rather than a journalist; otherwise, he would have single-handedly raised the proportion of abusive tweets to male journalists to almost 6 percent of the total.) While about three-quarters of the offenders were men, about 40 percent of the abusive tweets to women were sent by other women.

Are women really singled out for abuse on the Internet—or is it more the case that Internet abuse of women, at least when committed by men, is singled out for special concern and opprobrium? Many feminists would argue that there are good reasons to treat it differently. Sexual slurs toward women evoke the threat of real-life sexual violence; they are also perceived as intended to “put a woman in her place” and tell her that her opinion is worthless because she is a woman. (A sexual slur toward a man is considered just a personal insult.) But the double standard also has overtones of traditional chivalry which views women as more delicate and deserving of consideration—while nastiness toward men is treated simply a part of the rough-and-tumble of public life, to be taken in stride and shrugged off.

The Demos report on Twitter abuse toward public figures is only one of several studies that cast doubt on the assumption that the Internet is a hotbed of male hostility toward women. In an earlier Demos analysis, women on Twitter were almost as likely as men to use “gendered” derogatory language.

A survey of Internet users conducted by the Pew Research Center last year found that 13 percent of female respondents and 11 percent of male respondents said they had been harassed or stalked online. (While Hess’s Pacific Standard article drew on that survey to note that “5 percent of women who used the Internet said ‘something happened online’ that led them into ‘physical danger,’” it made no mention of the fact that 3 percent of the men also reported such an experience.)

Women and men also had very similar concerns about Internet privacy and security; while more women than men (77 vs. 61 percent) had changed their privacy settings to restrict some people from seeing their activities, women were only marginally more likely to have blocked or unfriended someone in the social media or to have asked someone to remove online information posted about them.

The most dramatic claim of pervasive online misogyny—that Internet accounts with feminine usernames get an average of 100 threatening or sexually abusive messages a day, to just four for male usernames—comes from a 2006 study conducted in Internet relay chatrooms, hardly typical of today’s social media use. Women also account for 72 percent of those reporting harassment incidents to the volunteer organization Working to Halt Online Abuse. But this is a self-selected sample, and the disparity may well reflect the fact that it’s more socially acceptable for women to seek help when they are harassed. (It should also be noted that the website’s annual rate of complaints from women is fewer than 300—obviously a cause for concern, but hardly an epidemic level.)

If women are unsafe or unwelcome on the Internet, it’s hardly borne out by the numbers. Women consistently use social networking sites more than men, outnumbering men on Facebook by about 10 percentage points; while there is conflicting evidence on whether Twitter has shifted from a majority-female demographic to a more male-skewed one, the likely reasons include the growing popularity of other platforms such as Pinterest and Tumblr along women.

Of course, none of this diminishes the deeply disturbing fact of female journalists, bloggers, and activists—Hess among them—who have been targets of threats to themselves or their families. But are men really immune from such attacks? One blogpost (civilly) critical of Sarkeesian and her supporters offers a fully sourced compilation of online comments wishing death, rape, mutilation and deadly diseases upon Jack Thompson, an activist critical of violent and sexual content in videogames—as well as death threats directed at male videogame developers who ran afoul of their fans. Meanwhile, role-playing game designer James Desborough claims to have been viciously threatened for defending the use of sexual violence as a plot element in games. And film blogger Alex Sandell (Juicy Cerebellum) has described receiving not only a deluge of hate mail but threatening phone calls—sometimes in the middle of the night, and sometimes made to his relatives—after writing negative reviews of the first two Lord of the Rings movies.

In the political sphere, several conservative male writers and activists have been targeted for rape and death threats, with their phone numbers publicly posted, after producing a documentary critical of the Occupy movement. Right-wing bloggers involved in the bizarre war with leftist activist Brett Kimberlin that David Weigel recently chronicled in The Daily Beast have faced scary cyber-harassment from some of Kimberlin’s supporters, including graphic fantasies of violent revenge, lurid sexual slurs, and accusations of child pornography. (Particularly disturbing examples are documented in a blogpost by First Amendment advocate Ken White.) On the other side, Charles Johnson, who runs the blog Little Green Footballs, relocated to a gated community because of threats he received after breaking with the right and embracing more liberal politics.

One male victim of cyberstalking, British expatriate novelist James Lasdun, told his story in the 2013 memoir, Give Me Everything You Have: On Being Stalked. Lasdun’s stalker, a former creative-writing student whose romantic overtures he had rejected, not only barraged him with abusive messages but emailed his colleagues accusing him of stealing her work, preying on female students, and even setting her up to be raped; she posted similar slanders on websites including Amazon.com and Wikipedia. But an experience like Lasdun’s gets no political sympathy; indeed, the review in The New Yorker chided him for failing to admit his “crush” on the woman and his role in leading her on.

Online abuse and trolling that doesn’t rise to the level of stalking or threats can still be toxic. When directed at women, such abuse can certainly take misogynist forms (whether because the troll actually hates women or because it’s the surest way to bait and provoke his—or her—target). But it can also come from feminists—both in internecine feminist wars, discussed recently by critics such as journalist Michelle Goldberg in The Nation, and in attacks on female dissenters. (Writer and blogger Susannah Breslin faced a particularly vicious backlash a few years ago after she criticized the then-new trend of “trigger warnings” in feminist media.) And plenty of time, such abuse is directed at men, in male-specific ways that we rarely think of as “gendered”: from questioning someone’s manhood to attacking his “male privilege,” from taunts about penis size to accusations of pedophilia, rape, or pro-rapist sympathies. The torrent of sometimes violent invective heaped last year at Dr. Phil McGraw after his tweet asking if it’s OK to have sex with a drunk girl was seen as righteous outrage against a “rape apologist” question, not online abuse.

Often, in our commitment to free speech, we have been too willing to accept toxic behavior on the Internet. Just because abusive speech is protected by the First Amendment doesn’t mean that private websites should give it a platform. There is, too, a very real need for law enforcement to catch up with technology and offer meaningful solutions to cyberstalking and threats. However, the war on online abuse, particularly with the agenda of making the Internet safe for women, has its own perils.

There is the obvious danger of censoring legitimate speech. In Canada right now, a middle-aged designer named Gregory Elliott is on trial for criminal harassment for sending non-sexual, non-threatening, but argumentative unwelcome tweets to feminist activist Stephanie Guthrie. But there is also the danger of perpetuating women’s vulnerabilities in the name of protecting them. In her analysis of videogames, Sarkeesian has been particularly critical of damsel-in-distress stereotypes and casual “depictions of female victimhood.” Yet we bolster the same stereotypes when we focus on nasty things said to women while trivializing threats against men even though men are much more likely to be victims of violence by strangers.

As long as the Internet exists, there will be rude, nasty, and unstable people on it—and sometimes, you will be attacked, especially if you write and speak on controversial subjects. We need a better middle ground between telling victims of harassment to grow a thick skin, and telling people they have a right to be shielded from all un-pleasantries. As we search for that middle ground, we should beware of paternalism based on the mistaken view that Internet nastiness is a particular problem for women.