Frank McCourt stepped in front of an 8-foot-tall screen, summoning a cluster of what looked like white, pixelated sand that hovered beside him. When he moved his hand over another screen, another cluster appeared; this time, the glittering particles took the shape he drew. Walking along the screen, he swept his arm wide to reveal its hidden message: “YOU CONTROL YOUR DATA.”

The...

The Future of Everything covers the innovation and technology transforming the way we live, work and play, with monthly issues on transportation, health, education and more. This month is Data, online starting Dec. 2 and in print Dec. 9.

Frank McCourt stepped in front of an 8-foot-tall screen, summoning a cluster of what looked like white, pixelated sand that hovered beside him. When he moved his hand over another screen, another cluster appeared; this time, the glittering particles took the shape he drew. Walking along the screen, he swept his arm wide to reveal its hidden message: “YOU CONTROL YOUR DATA.”

The billionaire real estate mogul was in Manhattan’s Hudson Yards area in September to show off an art exhibit he funded to demonstrate a new model for consumer data. Each digital dot represented one data point that social-media companies collect on him. The first screen depicted today’s model: Social media, search and online retail companies capitalize on information from consumers, who don’t own their data and see little return from the value they create with their clicks. Personal data is behind the $455.3 billion digital-ad market. The second screen represented Mr. McCourt’s vision, where individuals own their data and control how it’s sold and used by companies.

“An unprecedented amount of value is being created by everyone’s data. And yet there’s a total disconnect between the creation of that value and who receives it,” Mr. McCourt says.

Mr. McCourt is among a handful of business leaders, politicians and startups, such as Brave Software Inc., Reklaim Ltd. and Streamlytics Inc., seeking to empower consumers to control and benefit from the personal data that the technology industry collects on individuals every day. He has pledged $250 million to found Project Liberty, an initiative to rebuild the web as an infrastructure owned by the public. That includes $25 million to develop a decentralized social networking protocol, which he hopes will create a model for personal data governance online. Mr. McCourt, whose background is in real estate and sports ownership, says he was drawn to try to reform the tech industry because of what he sees as social media’s exploitation of users.

In his vision, blockchain software, not owned by any individual, would store indelible information about users’ social connections. After getting permission from users, social-media companies would draw from this same pool of data about users’ connections and interactions. Theoretically, this would allow users to move their data between networks and prevent a single company from becoming too powerful. Users could get paid in a cryptocurrency for the use of their data. For example, a user could choose to share posts within a small community that could make its own rules on monetizing data, moderating content and other factors. Eventually, apps for other online activities, such as shopping, might be built on the same technology, with users able to exercise the same control.

Users who make their data available to as many companies as possible might hypothetically make a few hundred dollars a year, says Braxton Woodham, president of Unfinished Labs, the arm of Mr. McCourt’s investing firm, McCourt Global Inc., behind Project Liberty. His pitch isn’t about promising users they’ll get rich, he says, but about avoiding exploitation. Mr. McCourt, who owned the Los Angeles Dodgers, started Project Liberty in 2020. He remains chairman and chief executive of McCourt Global.

“An unprecedented amount of value is being created by everyone’s data. And yet there’s a total disconnect between the creation of that value and who receives it.”

— Frank McCourt, founder of Project Liberty

The road to his vision is unclear. “The problem with these networks isn’t getting them to work” technologically, says Ethan Zuckerman, associate professor of public policy information and communication at the University of Massachusetts Amherst. The problem is convincing enough users to leave existing social networks, he says. Social-media companies optimize their networks to keep users there. Would a decentralized social network be as sticky? Plus, blockchain technology tends to make the web run more slowly and expensively, he says.

Politicians like California Gov. Gavin Newsom and former presidential and New York mayoral candidate Andrew Yang have floated the idea that consumers should get paid for their data. Some 60% of respondents to a Project Liberty poll of 1,022 U.S. adults in September said they would keep their data private if they owned and controlled it, while another 23% would exchange it for something of value.

Some startups are already putting this into practice. The browser Brave pays its 42 million users 70% of the revenue it generates from ads they see. Brave compensates them in its own “Basic Attention Tokens,” which they can redeem for currency or use to tip their favorite sites. Users report earning $5 to $10 monthly, according to a Brave spokeswoman.

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Foursquare Labs Inc., the social check-in app-turned-location-aggregator, launched an app in August that rewards consumers with gift cards for sharing their location. Tapestri Inc. introduced an app in November that pays consumers cash for sharing their anonymized location history, which the startup then sells to brands. Users can make up to $15 a month, according to Tapestri founder and Chief Executive Walter Harrison. Tapestri users can’t control which companies buy their data, but the company hopes to add that feature, Mr. Harrison says.

Reklaim, for example, gives users a view into the information that the industry buys and sells on them. When a user signs up for Reklaim and verifies her identity, the company uses her email address or phone number to query some 30,000 partners for data on that individual and then shows it to her. If users want to let Reklaim sell that data on their behalf, they get weekly checks as compensation. They can also complete surveys to add more data to their Reklaim profile and get paid more. Some consumers make up to $10 a month through the app.

Much of the data that consumers see about themselves in Reklaim is inaccurate, Chief Executive Neil Sweeney acknowledges, because it’s often the result of assumptions made about the consumer without direct verification. Consumers can correct this data, making it more valuable for brands who buy data from Reklaim.

Today, efforts to put consumers in control are limited. A world where they are compensated for their data is far off and beset with challenges, such as attracting users to new tools and upending the existing data-ownership market, which covers both entrenched social-media companies and data brokers that operate without transparency.

Some startups are taking advantage of privacy laws that require companies to make available to consumers all the data they have on them—a process that can be tedious and time-consuming for the user. Three-year-old startup Streamlytics has an app called Clture, which pays Black users for their data from sites like Netflix, YouTube, Amazon, Uber and Google Maps. Users can download their data from these sites, then upload it to Clture for cash. Streamlytics strips out the users’ personally identifying information and sells the data to businesses seeking insights into Black consumers’ habits. Streamlytics plans to launch additional apps focused on other communities.

“There are a lot of companies that make a lot of money off of the African-American community, particularly around different things going viral,” says Angela Benton, founder and chief executive of Streamlytics. “That never really gets funneled back into the community.”

Streamlytics decides how much an individual user’s data is worth using an algorithm that considers the stock performance and market capitalization of the company that the data comes from. Streamlytics’ payouts run as high as a few hundred dollars per quarter for users who maximize the service, Ms. Benton says.

“If you have a trillion-dollar company and a large part of that company is built on data inherently, the data that supports that company is worth more than, you know, a dollar or two dollars,” says Ms. Benton.

Write to Patience Haggin at patience.haggin@wsj.com