Some Facebook employees put out word of a "virtual walkout" to take place Monday to protest, according to tweeted messages.
"As allies, we must stand in the way of danger, not behind. I will be participating in today's virtual walkout in solidarity with the black community," tweeted Sara Zhang, one of the Facebook employees in the action.
Nearly all Facebook employees are working remotely due to the pandemic.
"We recognise the pain many of our people are feeling right now, especially our Black community," Facebook said in response to the AFP request for comment.
"We encourage employees to speak openly when they disagree with leadership."
Facebook was aware some workers planned the virtual walkout and did not plan to dock their pay.
"Mark is wrong, and I will endeavor in the loudest possible way to change his mind," Ryan Freitas, the design director of Facebook's News Feed, tweeted Sunday, adding that he was organising about 50 other employees who share his view.
Twitter changes display pic, bio to #BlackLivesMatter to protest George Floyd's killing
At the root of the discord is Twitter's unprecedented intervention last week when it tagged two Trump tweets about mail-in ballots with messages urging people to "get the facts."
Zuckerberg reacted by telling Fox News that private social media platforms "shouldn't be the arbiter of truth of everything that people say online." Trump retweeted the interview.
On Friday, Twitter responded once again to a Trump tweet, this time after he used the platform to warn protesters outraged by the death at police hands of an unarmed black man that "when the looting starts, the shooting starts."
Twitter covered up the tweet with a message warning it "violated Twitter Rules about glorifying violence." Viewers had to click on the message to see the underlying tweet.
The message also was posted on Facebook, but Zuckerberg decided to let it stand unchallenged.
"I've been struggling with how to respond to the President's tweets and posts all day," he wrote Friday in a post.
"Personally, I have a visceral negative reaction to this kind of divisive and inflammatory rhetoric."
But, Zuckerberg went on to say that "our position is that we should enable as much expression as possible unless it will cause an imminent risk of specific harms or dangers spelled out in clear policies."
- Network in revolt -
Twitter and Facebook both have in place systems to combat disinformation and dangerous content appeals to hatred, harassment, incitement to violence, and the like.
But Facebook exempts political personalities and candidates from these restrictions.
Zuckerberg's position has not gone down well with many of his employees.
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"I don't know what to do, but I know doing nothing is not acceptable," Jason Stirman, a member of Facebook's research and development team, wrote on Twitter.
Other Facebook employees spoke out on Sunday.
David Gillis, a member of the design team who specialises in product safety and integrity, said he believed Trump's looting and shooting tweet "encourages extra-judicial violence and racism."
"While I understand why we chose to stay squarely within the four corners of our violence and incitement policy, I think it would have been right for us to make a 'spirit of the policy' exception that took more context into account," he wrote.
Nate Butler, a Facebook product designer, added: "I need to be clear – FB is on the wrong side of this and I can't support their stance. Doing nothing isn't Being Bold. Many of us feel this way."
- A presidential call -
To make matters worse, US media revealed Sunday that Zuckerberg and Trump spoke by telephone on Friday.
The conversation was "productive," unnamed sources told the Axios news outlet and CNBC. Facebook would neither confirm nor deny the reports.
The call "destroys" the idea that Facebook is a "neutral arbiter," said Evelyn Douek, a researcher at Harvard Law School.
Like other experts, she questioned whether Facebook's new oversight board, formed last month to render independent judgments on content, will have the clout to intervene.
On Saturday, the board offered assurances it was aware there were "many significant issues related to online content" that people want it to consider.
Facebook, meanwhile, is directly affected by Trump's counter-attack against Twitter.
The president signed a decree Thursday attacking one of the legal pillars of the US internet, Section 230, which shields digital platforms from lawsuits linked to content posted by third parties while giving them the freedom to intervene as they please to police the exchanges.
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