The investment company says this oversight has contributed to Facebook "missing or mishandling" several "severe controversies" in past years, which they say increases risk exposure and costs to shareholders.
The specific examples the shareholders used include: Russian meddling in US election, the sharing of 87 million users' personal data with Cambridge Analytica, proliferating fake news and social unrest in Myanmar and Sri Lanka.
Chances of Zuckerberg's roles being split remain slim given Facebook has rejected similar shareholder requests in the past. A proposal last year to oust Zuckerberg as chairman received 51 per cent of the votes.
The push to split Zuckerberg's role into two comes as the social media giant continues to face concerns, including:
• Shares plummeting by some 21 per cent, wiping out an estimated US$150 billion in market value
• European privacy rules that went into effect in May, which partly resulted in Europe users dropping from 377 million to 376 million
• Fallout from Cambridge Analytica scandal in which personal data from 87 million Facebook users was collected, prompting several apologies from Zuckerberg
• Facebook's new video feature allowing Infowars on its platform, despite it peddling conspiracy theories including that Sandy Hook was a hoax
• Zuckerberg was also called on to remove harassing comments from Sandy Hook conspiracy theorists directed at the parents of victims
News of the proposal came just hours after Facebook shares went into a freefall on Wednesday as a stunningly weak financial outlook raised fresh concerns for the social networking giant.
After-hours trade saw Facebook shares plunge by some 21 per cent, wiping out an estimated US$150b in market value. If the share drop holds on Thursday, it would be Facebook's largest single-day decline, topping a 12 per cent decrease in July 2012.
Zuckerberg's personal wealth also took a hit with him losing nearly US$20b in just two hours. He saw his net worth tumble by US$18.8b, a record drop, in after-hours trading, taking him down four spots in Forbes' World Billionaires List.
He woke up as the fourth richest person in the world with an US$82.4b net worth on Wednesday, but was in the eighth spot by the end of the day.
Facebook's monthly active user count was 2.23 billion, slightly behind the 2.25 billion forecast by analysts. Users in Europe dropped from 377 million to 376 million, partly as a result of the new General Data Protection Regulation (GDPR) rules.
The second-quarter results were the first sign that the new European privacy laws and a succession of privacy scandals involving Cambridge Analytica and other app developers have bit into Facebook's business.
Facebook executives made investors even more nervous when they warned of further revenue deceleration in a call with analysts, citing efforts to address concerns about poor handling of users' privacy and to better monitor what users post.
CFO David Wehner said Facebook expects to see high-single digit drops in year-over-year revenue growth during the next few quarters.
He cited the new GDPR rules, user privacy controls and currency headwinds as factors contributing to the deceleration.
Wehner reiterated that GDPR didn't have a significant impact on revenues because it was implemented toward the tail-end of the quarter, but he said that could change in the coming quarters.
"We do think there will be a modest impact and I don't want to overplay these factors," Wehner said in the call with analysts.
"... We're continuing to focus our privacy model around putting privacy first. We believe that will have some impact on revenue growth. It's really a combination of how we're approaching privacy and GDPR and the like. All those factors together are... (things) we're considering."
The Cambridge Analytica scandal prompted several apologies from Zuckerberg and generated calls for users to desert Facebook, which has grown strongly since launching as a public company in 2012.
Ad sales in Facebook's second quarter rose 42 per cent to US$13.04b but the costs, bolstered by moves to improve content and security after the data scandal, rose 50 percent from a year earlier to US$7.37b.
Total revenue rose 41.9 per cent to US$13.23b, while Wall Street was looking for revenue of US$13.36b.
Ahead of the announcement, industry experts had predicted that the number of active users visiting the social network would either drop or flat line.
Facebook has never reported anything other than user growth in Europe.
Still, Zuckerberg assured investors that Facebook continues to see growth on its core platform, as well as its other properties, which include Instagram, WhatsApp and Messenger.
For the first time ever, Zuckerberg provided user growth data on those properties, saying that Facebook now has 2.5 billion people who use at least one of its apps, referring to users who may have an Instagram account, as well as a Facebook profile.
"Our community and business continue to grow quickly," he said. "We are committed to investing to keep people safe and secure, and to keep building meaningful new ways to help people connect."
According to analysts, stalling numbers are likely the result of forcing Facebook users to consciously opt-in to having their information used for personal advertising - one of the key stipulations of the stringent GDPR rules.
GDPR, which came into affect May 25, stipulates that companies must explicitly request consent from their users in order to personal data for advertising purposes.
Companies that do not comply with GDPR can be fined up to four per cent of their global revenue.
To comply with the new regulations, Facebook rolled-out a security check-up to users worldwide which asked them to review what kind of personal information they consent to sharing for advertising targeting. Users were also asked to consent to facial recognition technology on the site.
Analysts believe users may have been scared off by the explicit details about how their data is being used by the social network.
Meanwhile, in addition to the financial woes, a promotional event for Facebook's new video feature descended into a shouting match on Wednesday, as reporters quizzed the social media giant's executives about its decision to allow Fox News and Infowars on the platform.
Facebook will be featuring Fox News alongside other news organizations on Watch, while right-wing conspiracy theory site Infowars has a Facebook page that is followed by nearly a million people.
The debate at the Television Critics Association's press tour in Beverly Hills began when Facebook's head of video was asked why Facebook still hosted Infowars when it peddled conspiracy theories including calling the Sandy Hook massacre a hoax.
The reporter noted that Infowars host Alex Jones recently spread a fake story about special counsel Robert Mueller being a pedophile, telling his audience: "They'd let Mueller rape kids in front of people, which he did."
Fidji Simo replied that although he found Infowars "absolutely atrocious", Facebook's approach to freedom of expression meant the company would only reduce the distribution of Infowars posts, rather than ban it completely.
Zuckerberg was also being called on to remove hateful and harassing comments from the social media platform that have been posted by conspiracy theorists who don't believe the Sandy Hook Elementary School shooting happened.
Leonard Pozner and Veronique De La Rosa, the grieving parents of one of the 20 children killed in the 2012 Newtown shooting massacre, wrote Zuckerberg a letter published on Wednesday in The Guardian.
Pozner and De La Rosa say they and other relatives of mass shooting victims have been harassed and threatened on social media and in person by people who claim the shootings were government hoaxes and the victims - including their son Noah - were actors.
"Our families are in danger as a direct result of the hundreds of thousands of people who see and believe the lies and hate speech, which you have decided should be protected," they wrote.
"We are unable to properly grieve for our baby or move on with our lives because you, arguably the most powerful man on the planet, have deemed that the attacks on us are immaterial, that providing assistance in removing threats is too cumbersome, and that our lives are less important than providing a safe haven for hate."
Other parents of Sandy Hook victims and people affected by other mass shootings, including the one that killed 17 people at a Florida high school, have made similar requests of Facebook, Twitter and other social media companies.
Zuckerberg raised eyebrows in an interview with Recode last week when he said he finds denial of the Holocaust "deeply offensive" but doesn't believe such content should be banned from Facebook.
"What we will do," he said, "is we'll say, 'Okay, you have your page, and if you're not trying to organise harm against someone, or attacking someone, then you can put up that content on your page, even if people might disagree with it or find it offensive.'"
He also said in the interview that the claim the Sandy Hook shooting didn't happen is false.
"I also think that going to someone who is a victim of Sandy Hook and telling them, 'Hey, no, you're a liar' - that is harassment, and we actually will take that down," Zuckerberg said.
In a statement Wednesday in response to the letter by Pozner and De La Rosa, a Facebook spokeswoman said the company recognizes that victims of mass shootings and other tragedies are "vulnerable to offensive and incendiary comments."
"Although we do see people come together on Facebook in very positive ways around tragedies, some of what we see is truly abhorrent and represents the worst of the internet and humanity," the statement said.
"We don't allow people to mock, harass or bully the victims of tragedies. This includes the types of claims in the letter that victims are crisis actors. We also don't allow people to celebrate, justify or defend the tragedy in any way."
