The balance in the art world is shifting. As the world’s richest private collectors get richer, publicly financed museums are, by and large, getting poorer.

Last week, critics made their first assessments of The Broad, the 120,000-square-foot private museum in downtown Los Angeles to showcase the contemporary art collection of the billionaire philanthropists Eli and Edythe Broad. The museum has an endowment of more than $200 million, and additional funds for acquisitions.

Also last week, four other collectors were displaying their art in a very different showcase that could be seen as an alternative model for private collectors. Four museums and a church in Sheffield, a town in Yorkshire in northern England, famed in Victorian times for its steel-making, opened their own exhibition of contemporary art, running through Dec. 12.

Squeezed by the British government’s austerity program, Sheffield City Council now finances its local museums to the tune of 1.5 million pounds a year, about $2.3 million, a reduction of more than 30 percent since 2009. The acquisition fund stands at just 70,000. That doesn’t go very far in today’s art market, and any purchases by British regional museums have to be supported by financing from public bodies like Arts Council England and private donations. A show supported by private collectors, therefore, can be a boon to the museums, and to the collectors themselves.

“Private collectors have a lot of money and buying power that has driven growth in the art market,” the Berlin art adviser and writer Marta Gnyp said in a phone interview. Her book on the subject, “The Shift: Art and the Rise to Power of Contemporary Collectors,” was published on Aug. 4.

“Public museums have financial restraints,” she added. “But they are still attractive to private collectors. Public institutions give a quality stamp and visibility to collections.”

Museums Sheffield is using the exhibition project, called “Going Public,” and a conference at Sheffield Hallam University on Oct. 12 to examine how cash-strapped public galleries and wealthy private collectors can cooperate more effectively.

“Going Public” is the first time that four major international collectors of contemporary art have collaborated on a regional museum show in Britain. As part of the joint show, recent Chinese video and digital art from Dominique and Sylvain Levy’s DSL collection in Paris are on display at Sheffield’s SIA and Site galleries. At the Millennium Gallery, eight works of Minimalism — including the classic 1967 Sol LeWitt white-painted aluminum sculpture “Serial Project #1, A6” — have been lent by the French collector Nicolas Cattelain in London. Marcel Duchamp’s 1952 “La Boite-en-valise” (Box in a Suitcase) is among 35 Surrealist works being shown by the Berlin collector Egidio Marzona in the Art Deco Graves Gallery museum.

And unsettling installations — like a 1996 Chapman Brothers corpse hanging upside down over a vat of blood and a 2009 tapestry by the Polish artist Goshka Macuga of London foretelling today’s migrant crisis — are transforming the medieval interior of Sheffield Cathedral, courtesy of Patrizia Sandretto Re Rebaudengo, who has a private museum in Turin.

“They’re all true believers,” Sebastien Montabonel, an art consultant in London, said of the collectors who contributed work to the shows. He originated the project four years ago with Mark Doyle of the Contemporary Art Society in London. “They understand art is getting expensive and that their role is important.” He added: “Two or three collectors can make a huge impact, and being a big fish in a small pond is a good selling point.”

These selections from four serious collectors will give residents of and visitors to Sheffield a distinctive overview of some of the more interesting things international artists have been doing over the past 100 years. But can this model be repeated at other regional museums?

In February 2014, Mr. Marzona, one of the participating collectors in “Going Public,” transferred ownership of 372 pieces of 20th-century avant-garde art to the city of Berlin, having already given a substantial proportion of his collection to the city’s museums in 2002.

But nowadays, as Ms. Gnyp notes in “The Shift,” the private museum owned by a collector or an associated foundation has become the model, rather than donations to public institutions, despite the tax advantages that can accrue from such gifts. Ms. Gnyp writes that no fewer than 125 museums were created worldwide from 2006 to 2013, as opposed to just 25 in the 1990s. Wealthy individuals like Mr. Broad and Ms. Sandretto Re Rebaudengo want to keep control of their collections and play a more proactive role in the art world.

“I want to show my collection and share it with other people,” said Ms. Sandretto Re Rebaudengo, who began buying art in 1992 and now has more than 1,000 works. “But private collectors can do so much more. They can be involved in the production of art and in education.” For example, for the 2009 Venice Biennale she commissioned the Macuga tapestry hanging in Sheffield Cathedral. She said that “Going Public,” with its expansive use of loans, could prove a model for future collaborations between art’s public and private sectors.

But as private museums have grown in power and influence, so too has the fixation on contemporary art as an investment. Media reports of spectacular price rises for fashionable artists have attracted droves of newcomers to the market.

“There are probably no more than 200 or 300 real collectors who have a vision and have proper curation,” Mr. Montabonel said. “The others are just buyers. Their ‘collections’ will be forgotten in 10 years’ time.”

Is it that simple, though? Are there such clear distinctions between “real” collectors and mere “buyers”?

The Broads were collecting contemporary art long before it was being promoted in power point presentations as an “alternative asset class.” Yet a number of visitors to their gleaming new museum have noted how the procession of works by blue-chip names like Jeff Koons, Ed Ruscha, Mark Bradford and Mark Grotjahn hung on temporary white partition walls can make the display, at times, resemble the world’s most expensive art fair.

“The lack of independent taste among high-powered collectors is most depressing,” Kenneth Baker, the former art critic at The San Francisco Chronicle, said in an email. He has viewed the Broad collection twice but has yet to visit the new museum. “Broad and his ilk certainly don’t have to think like speculators when they shop for art, unless they can’t help themselves.”

The difference could not be greater when contrasted with the subject of the first exhibition at Damien Hirst’s private museum, which will open in a 37,000 square-foot converted factory in Lambeth in South London on Oct. 8. The inaugural show will focus on the Sheffield-born abstract painter John Hoyland (1934-2011), a surprising choice for the former superstar of Young British Art.

Now 50, Mr. Hirst is ranked as the 450th wealthiest person in Britain, worth 215 million, according to The Sunday Times’s “Rich List” for 2015. The artist has been a compulsive buyer since the 1980s, and his Murderme Collection now contains more than 3,000 works by Picasso, Warhol, Banksy, Bacon and a host of others. As it turns out, he is also an admirer of Hoyland, who was influenced by the Abstract Expressionists, and whom he discovered when visiting Leeds Art Gallery as a schoolboy.

“In my eyes, John Hoyland was an artist who was never afraid to push the boundaries,” Mr. Hirst says on his website.

Unlike the opening of The Broad, the first Hirst Museum show might be closer to Mr. Montabonel’s notions of “true” collecting. As yet, Hoyland doesn’t command much of an international market. The highest auction price for one of his works is “just” 185,000, paid back in 2008. By and large, people looking at these paintings won’t be distracted by their monetary worth, and the experience will be closer to that of visiting a public museum.

The “financialization” of contemporary art, and the ways this has, or has not, influenced collectors, might be an interesting break-out topic for that Sheffield conference in October.

In the meantime, what do Yorkshire’s down-to-earth locals think of this fancy international art dropping into their city?

“I like it,” Ronald Fletcher, 87, said outside Sheffield Cathedral. “I’m not overly religious. I don’t really believe in God, but I’ve been looking for something since I lost my wife. I find it refreshing.”

At least someone knows the true value of art.