Restaurant managers and hosts are being assigned a new job—vaccination police—and it’s dividing businesses and customers.

Some U.S. municipalities are asking restaurants, bars and other businesses to verify patrons’ Covid-19 vaccination status before they can dine or drink indoors, much as some European nations have begun to do. The local mandates come as some bars and restaurants have voluntarily started asking patrons to prove their vaccination status, including a growing number in Seattle, San Francisco and Los Angeles.

Some restaurant owners checking diners’ vaccination status or a negative Covid-19 test say the practices—intended to help move beyond the pandemic—introduce new logistics.

Customers have canceled reservations and gotten angry when showing up unaware of the new rules, some owners said. Some online reviews of their restaurants now blast the new policies. Other owners say they and their staff have received anonymous threats from people who accuse them of infringing on individual freedoms and discriminating against the unvaccinated.

“People scream at us through the phone. They say racist things. It’s all about the vaccines,” said James Lim, owner of the Watson’s Counter restaurant and cafe in the Seattle suburb of Ballard, Wash.

Watson’s Counter owner James Lim says the staff supported the voluntary move to require indoor diners to show proof of vaccination.

Mr. Lim said his staff supported voluntarily instituting a proof-of-vaccination requirement when the restaurant reopened for dine-in service earlier this month. Many customers of the restaurant welcomed the move, he said, but the hostility has troubled Watson’s younger workers.

Thousands of other U.S. restaurants soon will be joining Watson’s. Starting Monday, New York City restaurants must verify that a customer has had at least one dose of a vaccine before entering to eat, with city enforcement starting on Sept. 13. New Orleans also will start to impose a vaccination mandate or negative Covid-19 test for indoor dining starting Monday. Philadelphia last week added an incentive for restaurants to check vaccination status, saying that indoor dining without masks can continue only if all patrons and workers are vaccinated.

San Francisco will begin enforcing one of the strictest rules Friday, allowing only fully vaccinated people to dine indoors. Elsewhere in California, Palm Springs and Cathedral City will begin requiring proof of vaccination or a recent negative Covid-19 test for indoor dining and drinking in the coming weeks. Los Angeles is weighing similar measures.

“Our economy is so dependent on tourism. It’s essential to be known that we are safe,” said Palm Springs Mayor Christy Gilbert Holstege.

The proof-of-vaccine screening is the latest in a series of pandemic-related requirements placed on restaurants over the last 18 months.

Many Watson’s Counter customers welcomed the move to require proof of vaccination, its owner says, but the restaurant also has been on the receiving end of hostility.

Some restaurants already requiring vaccination proof say they are willing to do whatever it takes to avoid their dining rooms being closed down again due to coronavirus surges. Owners say they believe loyal customers will reward them with business and they are willing to endure the criticism.

“We don’t want to get shut down again. It’s that simple,” said Dean Lavine, owner of Palm Springs bar and restaurant Blackbook, which started checking customers’ vaccination status earlier this month.

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Others are skeptical about private businesses being asked to enforce vaccine mandates. Some fear driving unvaccinated people away, or confusing customers.

Laurie Thomas, owner of two San Francisco establishments and executive director of the Golden Gate Restaurant Association, said she supports the city’s new rules, but worries about her staff having to enforce the mandate.

“Tell me the last time you were carded at a restaurant? It’s a different level of training,” said Ms. Thomas, who wants the mandate to be widely publicized so customers aren’t surprised.

Some restaurants requiring proof of vaccination say they are willing to do whatever it takes to avoid another shutdown.

Business for restaurants was strong heading into the summer but has slipped as the Covid-19 Delta variant has spread. Restaurant sales growth in the week ended Aug. 1 was the slowest in nearly two months, according to restaurant-analytics firm Black Box Intelligence. More customers are mentioning Covid-19 safety practices in recent online reviews, Black Box found.

Americans are split over needing to disclose vaccination status to dine out.

Janet Russell, a property manager from Arlington, Va, said she’s opposed to rules requiring proof of vaccination. “These small businesses that have already suffered tremendously during lockdown should not have to shoulder the burden,” Ms. Russell said.

Restaurateurs are divided, too. Palm Springs businesses largely supported the move before it was adopted, for example, while most Colorado restaurants are opposed to proof-of-vaccine mandates out of concern for their staff having to police customer compliance, according to polls by local business groups. Some restaurant managers, fearing lawsuits from disgruntled patrons, said they want clear guidelines from local officials on how to determine vaccination status.

In New York City, restaurant operators are trying to determine how their workers will verify customers’ vaccination status. The city order also requires all restaurant workers to be vaccinated.

Executives at Shake Shack Inc., Panera Bread Co. and Applebee’s-parent Dine Brands Global Inc. all said they intended to comply but are still trying to understand the requirements.

“I do worry a little bit about the responsibility of checking ID cards, and the place that puts our hosts and hostesses,” said Dine Chief Executive John Peyton. “I’m hopeful that New Yorkers are respectful of what’s now required.”

Surveys have found owners and consumers to be divided over vaccine mandates for restaurants.

A survey of 536 consumers last month by market-research firm Datassential found diners about evenly split over whether they would comply with a proof-of-vaccination requirement at a restaurant.

“I understand the objections to it. People think it’s unfair,” said Sam West, a 26-year-old Manhattan restaurant server. “But now I think of it as a necessary evil.”

South Seattle small-plates restaurant Off Alley began requiring diners to show proof of vaccination in late July. Co-owner Meghna Prakash said customers have generally been supportive but online backlash is taking a toll. Off Alley has received negative online reviews from people who haven’t dined there, and she has been called discriminatory, Ms. Prakash said.

“Making these policy decisions by ourselves is putting targets on our backs,” she said.

Write to Heather Haddon at heather.haddon@wsj.com