“The idea was that, unlike a traditional home swap, you didn’t have to go to the same person’s house who comes to yours because you may not want to take your vacation there,” Mr. Shealy said. “You had the opportunity to go to a place you really wanted.”

The network quickly grew, and today, Thirdhome has a portfolio of almost 11,000 luxury properties in 95 countries and does between 600 and 700 exchanges a month.

Homeowners must apply to have their house listed and are required to give it up for a minimum of one week a year. The property cannot be their primary residence and must meet a list of criteria to be accepted: It needs to be in an appealing location for that destination, for one. A home in the Caribbean, for example, such as Villa Celeste, should be near or on the beach while a property in the Swiss Alps should be close to skiing. The homes also need to have top-of-the-line kitchen appliances, up-to-date bathrooms and high-end furnishings; Mr. Shealy said that a member of Thirdhome’s business development team vets the properties, either in person or through local brokers.

Although the average home listed is valued at $2.3 million, some are more extravagant than others.

Great Wall Villa in Beijing, for example, is a $6 million, 8,000-square-foot spread built into a mountain overlooking the Great Wall and surrounded by rock formations and forest. It has eight bedrooms and comes with a two-person staff.

Then there’s the 5,500-square-foot ultramodern home, also worth $6 million, in Queenstown, New Zealand, with a helipad and views of Lake Wanaka, regarded as one of the country’s most spectacular sites. The property is near skiing, hiking and golf.

Thirdhome also has several thousand hotel residence listings including ones at Ritz-Carlton, Rosewood and Auberge properties globally.