In the United States, even before the Orlando massacre spawned new dread about “lone wolf” terrorism, a majority of American governors said they wanted to block Syrian refugees because some could be dangerous. Donald J. Trump, the presumptive Republican presidential nominee, has called for temporary bans on all Muslims from entering the country and recently warned that Syrian refugees would cause “big problems in the future.” The Obama administration promised to take in 10,000 Syrians by Sept. 30 but has so far admitted about half that many.

Just across the border, however, the Canadian government can barely keep up with the demand to welcome them. Many volunteers felt called to action by the photograph of Alan Kurdi, the Syrian toddler whose body washed up last fall on a Turkish beach. He had only a slight connection to Canada — his aunt lived near Vancouver — but his death caused recrimination so strong it helped elect an idealistic, refugee-friendly prime minister, Justin Trudeau.

The Toronto Star greeted the first planeload by splashing “Welcome to Canada” in English and Arabic across its front page. Eager sponsors toured local Middle Eastern supermarkets to learn what to buy and cook and used a toll-free hotline for instant Arabic translation. Impatient would-be sponsors — “an angry mob of do-gooders,” The Star called them — have been seeking more families. The new government committed to taking in 25,000 Syrian refugees and then raised the total by tens of thousands.

“I can’t provide refugees fast enough for all the Canadians who want to sponsor them,” John McCallum, the country’s immigration minister, said in an interview.

In the ideal version of private sponsorship, the groups become concierges and surrogate family members who help integrate the outsiders, called “New Canadians.” The hope is that the Syrians will form bonds with those unlike them, from openly gay sponsors to business owners who will help them find jobs to lifelong residents who will take them skating and canoeing. Ms. McLorg’s group of neighbors and friends includes doctors, economists, a lawyer, an artist, teachers and a bookkeeper.

Advocates for sponsorship believe that private citizens can achieve more than the government alone, raising the number of refugees admitted, guiding newcomers more effectively and potentially helping solve the puzzle of how best to resettle Muslims in Western countries. Some advocates even talk about extending the Canadian system across the globe. (Slightly fewer than half of the Syrian refugees who recently arrived in Canada have private sponsors, including some deemed particularly vulnerable who get additional public funds. The rest are resettled by the government.)

The fear is that all of this effort could end badly, with the Canadians looking naïve in more ways than one.