There was a Cybertruck parked on Main Street in East Hampton, outside the Altuzarra store. It was a Sunday afternoon in June, and traffic stalled for a moment. Even the rich are not immune to rubbernecking a brutalist behemoth.

The monster truck marked the end of an avenue of monograms — the island’s main luxury shopping drag, with $850 raffia handbags and $15,000 decorative surfboards. You know their names: Louis Vuitton, Loewe, Lululemon.

Two and a half miles down this same street, however, quaintness emerged. East Hampton turned into Amagansett, and that flashy boutique strip became a town square with white wood-paneled cottages. There was a shoe store called Brunch, a children’s clothing chain called Pink Chicken, a jewelry and gift shop called Love Adorned. A Cybertruck here would read as a declaration of war.

It was near these cottages that the Row, a brand founded in 2006 by Ashley and Mary-Kate Olsen, quietly opened a store on Memorial Day weekend.

Quietly is how the Row tends to operate. Not only in its clothing — often described as “quiet luxury,” a term used to describe very expensive basics — but also in its communication.

The founders rarely give interviews, advertise or otherwise promote their line. While the Row did announce its Amagansett opening on Instagram, that account is more outwardly devoted to sharing modern art than to moving product. In February, the brand caused a stir at Paris Fashion Week by asking its runway show attendees to “refrain from capturing or sharing any content during your experience” — which is, for many, the primary reason for attending a fashion show. The audience was encouraged to write down thoughts instead.

Somehow this stance works. In an industry overrun by influencers, the Row’s silence is stark. Monasticism is chic. There is an impression of exclusivity and taste, buoyed by the extreme prices. One of the Row’s most popular items, the Margaux bag, ranges in price from $3,490 to $6,810, depending on size and material. It is timeless and ladylike, the kind of purse that might remind Kendall Jenner of Jacqueline Kennedy Onassis.

The Row’s stores also have a reputation for being intimidating at times, even among seasoned high-end shoppers.

One loyal Row customer told me she felt like “peasantry” in the Los Angeles store, which houses an untouchable swimming pool. At the store in Manhattan — a townhouse with a limestone spiral staircase — “there is one guy who works there that all my friends are afraid of, who radiates a very ‘you can’t sit here’ vibe,” said Jess Graves, the writer of a shopping newsletter called The Love List, “even to girls I know who walk in wearing the brand head to toe.”

The Amagansett shop is different. It operates out of a house with roots in the 19th century, formerly occupied by Tiina the Store, the Hamptons’ Gap for billionaires. (Tiina stocked the Row.)

It has a porch and a screen door and a woven beige carpet. The fitting rooms are harshly lit behind denim patchwork curtains. (By contrast, the spacious wood-floored dressing rooms at the Upper East Side store, where I recently tried on a $1,550 white cotton poplin tent dress that made me look, tragically, like a hospital patient, have soft lighting and softer robes.)

There is no statement artwork in Amagansett, unlike the London store, where an oval light installation by James Turrell greets visitors at the entrance. The vintage furniture is noteworthy — there’s a black chaise shaped like a person from the 1970s by Olivier Mourgue Bouloum and a white painted wooden lounge chair from the 1930s by Robert Mallet-Stevens. But the décor, with its Asian and African influences, is not the point.

The point of the store is the large selection of jewelry, home wares, snacks and skin care by more than 20 brands and artisans that are not the Row. Shampoo from Florence. Beaded necklaces from Greece. A mother-of-pearl caviar set. A bronze lighter carved to resemble tree bark. A packet of dried mango and a jar of raw almonds. Vintage glass candlesticks that can be purchased only in a set of a dozen for $16,000.

There are racks of ready-to-wear clothes made by the Row, of course, the selection tailored to this beach town: bike shorts ($1,050), denim shirts (also $1,050), ribbed tank tops ($670), sleeveless silk maxi-dresses ($1,890). Ms. Graves bought herself a raffia bag here earlier in the season. (“It felt very appropriate while I’m out here this summer,” she said.)

But the Row confirmed that the Amagansett store is its first attempt at a “local” store concept. What this presumably means is a space that is more relaxed, filled with objects that complement the brand’s vision of itself, staffed by sales associates who do not scare people away but warmly help shoppers track down sold-out jelly flats. Not that the Row’s fans are easily scared away: Even those who are intimidated don’t stay away for long, these masochists for cream-colored cashmere.

In retrospect, the popular jelly shoes, along with the beach towels that models wore as scarves on the Row’s runway in September, may have been a sign that the brand was loosening up — that brightness and humor were coming to this austere world. (Its most recent look book showed a silky camisole dress layered over pants, Y2K-style.)

A British client of the Row visiting the Amagansett store marveled at the vibe shift. Where was the icy indifference? “I don’t think it would fly with the audience here,” she said.