Nude bathing for men was an accepted tradition in Victorian times. (A stretch of river called Parson’s Pleasure was reserved for this purpose at Oxford University until 1991.) But as “mixed” bathing became popular near the end of the 19th century, both men and women were expected to cover up from breastbone to knee.
1920’s bathing suits were clinging, but very similar for both sexes.
Practices varied from place to place but, at public beaches and pools in the U.S., men were usually required to wear suits that covered their nipples until the mid-nineteen thirties.
The “Speed Suit” (left) has attached trunks and “elastic-ribbed fabric.” The “High Waisted Trunks” at center are shown with a separate all-wool shirt which tucks into the suit at front and back. The “two-purpose” Zip Top Suit” at right has a zipper in front that allows you to remove the “shirt” part.
By 1934, it was becoming acceptable for men to swim bare-chested, but rules for public and private beaches and pools differed, so bringing an optional top would save embarrassment. (Speaking of embarrassment, I wonder: when the trunks were not suspended from the shoulders, was a belt necessary to support the weight of water-logged wool knit trunks?)
This vintage suit, from Macy’s, has a similar zipper front and a rather bare X back:
This illustration from Womans’ Home Companion, 1937, shows that some men — in this case, two out of three — continued to wear the top even when not required to do so.
The older man is wearing a more conservative, covered-up swimsuit.
According to Esquire magazine in 1934,
This implies that shirtless swimming was permitted on some public beaches in 1933, and earlier  at some private beaches and pools.
The man at left is wearing a shirt tucked into his trunks.
In the same July 1934 issue, this ad for Mansco Sportswear shows several conservative looks:
However, this ad from Gantner and Mattern Co. shows much tighter-fitting trunks — and no top.
The Wikies’ high waist reflects the high-waisted men’s trousers then in fashion. Wikies’ snug fit was probably possible because of the recent  invention of Lastex yarn, which even appeared in men’s suit fabric in 1934 ads.
The Lastex company ran a series of advertisements in Esquire magazine showing men’s suits, tuxedos, etc. which were made with stretch fabrics — in 1934!
This editorial illustration appeared in a women’s magazine in 1935:
Perhaps the acceptance of bare chests had something to do with Hawaii:
I can’t help noticing that Esquire chose to use men “of a certain age” to model swimsuits in its editorial fashion articles. The women’s magazines, however, pictured younger, athletic-looking men wearing swimsuits in their illustrations, just as Esquire favored voluptuous women in its cartoons….