Eli Broad’s recent announcement that he would not donate his art collection to the Los Angeles County Museum of Art, but would leave it instead with a private foundation, drew responses that ranged from howls to raves. “LACMA got screwed,” one prominent art critic said bluntly.

“Congratulations,” the former director of the Metropolitan Museum of Art, Thomas Hoving, wrote in an open letter to the Broads in the Los Angeles Times, “for your dynamic, imaginative, innovative decision.”

The interests of museums and their wealthy donors often exist in a state of dynamic tension. This is never more true than in the matter of how donors dispose of their art collections. A donor wants his collection to stay together; a museum would prefer to select the pieces that complement its holdings. Some donors want assurance that their art will be always on view. Others want to name the collection’s curator, or to hold curatorial powers themselves.

At the moment, a number of factors, including the immense accumulation of wealth in the last decade and the boom in contemporary art prices, have contributed to an energetic debate about the best way for collectors to simultaneously secure their legacies while serving the public interest.

Mr. Broad believes that the way museums operate is dysfunctional. “Museums have gotten into this practice of accumulating a huge amount of work and showing very little, while paying the costs of storing and conserving the work,” he said in an interview. “As time goes on, the practices of museums, which might have been fine 50 or 100 years ago, maybe ought to change.”

Mr. Broad believes that his model — the private foundation that operates as a lending library to museums and other educational institutions — represents a significant new paradigm and will attract imitators. He said he had already heard from some collectors who were interested in contributing their collections to his foundation — just like Warren Buffett chose to add his billions to the Bill & Melinda Gates Foundation, he noted.

“We weren’t expecting anything like that, but we’ve had a couple of calls,” he said.

Other collectors are following other models. The Gap founder, Don Fisher, after deciding not to donate his collection to either the San Francisco Museum of Modern Art or the de Young Museum, now plans to build his own 100,000-square-foot museum in the Presidio — that is, unless a consortium of groups that oppose the plan, including the National Park Service and the Sierra Club, manage to stop him.

Some others are pushing the idea of joint ownership among museums. Last year, Mr. Broad brokered a deal whereby LACMA and the Museum of Contemporary Art, Los Angeles, jointly acquired a large-scale sculpture by Chris Burden. More recently, the art dealer Anthony d’Offay sold a collection of some 750 works, at a discount, to the Tate Gallery and the National Galleries of Scotland; the works will be shown around the United Kingdom.

Apparent in these debates is that collectors and museum leaders have vastly different views about how well the current system works, as well as about what best serves the public good.

Mr. Broad’s observation that museums have only a small amount of their collections on view is accurate. Most museums exhibit only 5% to 10%, or even less, of their collection at any given time. Granted, there are some asterisks associated with these numbers: Many works, including works on paper and textiles, have to be stored a certain percentage of the time because they are light-sensitive. The Met, for example, says that this is true of nearly half its collection.

But museums also argue that their collections serve the public in ways beyond just being seen in the galleries. They are the subject of ongoing research by museum curators, as well as outside scholars, who can generally apply to have access to specific works for the purpose of their dissertations or books.

Museums also make loans to other institutions, a fact that Mr. Broad seems to ignore. The director of the Dallas Museum of Art, John Lane, described a recent case in which a couple, Nona and Richard Barrett, wanted to donate their collection of Texan artists to the Dallas Museum of Art and the Museum of Fine Arts, Houston. The MFA, Houston, chose the pieces it wanted for its collection. The Dallas Museum of Art chose the pieces it wanted, and agreed, as well, to accept the balance of the collection, on the understanding that the museum would put those works on longterm loan to other, smaller institutions in Texas.

“I think that was a creative way around the issue,” Mr. Lane said. In recent years, museums have taken many approaches to getting more of their collections on view. That has been a major impetus behind physical expansions. It is also one motive to establish satellites, as the Guggenheim has done, which bring the museum’s collection before more people and build its reputation. It’s hard not to see these debates boiling down to issues of whose image and name gets burnished — an individual’s or an institution’s. LACMA may not miss the long-term storage and conservation costs associated with acquiring Mr. Broad’s collection. But what it really loses is the prestige that would have come with the acquisition.

“Having a great collection is a magnet for the best curatorial talent and a motive for other people to give” their collections, Mr. Lane said. Having great works to loan also provides leverage to secure loans from other institutions. “When work stays in a private foundation it doesn’t have that empowering impact on an institution,” Mr. Lane said.

People in the museum world say there is a value to works being in a museum collection over the long term, where they can be compared and contrasted with other works in the collection, and combined and recombined in pursuit of different art historical narratives.

The question, then, is whether museums can persuade donors of the value of having their works seen in the context of other artists’ work that they did not collect. “From a donor perspective, why should I give you my Brice Marden painting — why should it sit and not be seen by people — when I can put it up in my suburban or exurban space where anyone who comes can see it?” the undersecretary for art at the Smithsonian Institution, Ned Rifkin, said. But a museum exhibition can pose the question: “Does Brice Marden hold up against Mark Rothko?”

Mr. Lane summarized the response in the art world to Mr. Broad’s announcement as a discussion of “questions of private ownership versus public welfare.” There is a longing, he said, “for works to become part of the public domain.”

But the fact that major collectors want to set up private institutions is not new. That’s how we got the Morgan Library, the Frick Collection, the Sterling and Francine Clark Art Institute, and many more. It’s also how we got the Barnes Foundation, whose longtime — and at this point, likely successful — effort to move to Philadelphia from suburban Merion illustrates some of the factors to be considered in establishing private institutions. Its founder, Albert Barnes, stipulated in its charter and bylaws that its collection not be moved, but the Barnes’s current board argues that the Foundation can neither remain financially solvent, nor serve its public mission well, if it stays in Merion.

The question is whether those individuals who are currently setting up private museums and foundations foresee all the challenges involved with keeping them functioning well in the long run.

“As a lot of people figure out, it’s really complicated to run an art museum,” Mr. Lane said.