I'm most familiar with New York City, which had a thriving gay community in the 1920s (and even before), so I will keep my answer to it and hopefully someone else can tell you how it compared to Britain! Since it is well documented in existing literature, the story of it's culture should be able to give you an overview of the way things were.

Rooming houses were common in big cities like NYC and Chicago because they were affordable housing options for single, working class folks. They were often old, run down hotels where a person could rent a room by the week or month to have a place of their own to keep their belongings and to sleep when they weren't working.

By no means were all tenants of rooming houses gay, but due to the fact that gay men were often unmarried, it makes sense that this kind of housing would become a bit of a magnet for them. Through living (and usually eating, more on that in a moment) in the same place, those gay men would eventually find one another and build friendship networks and small local communities. These networks and communities in cities were probably the closest thing to support groups that existed before the Mattachine Society was formed in 1950 (though it wasn't exactly a support group, but more on that in a moment as well).

I know that sounds oversimplified, but it really was that simple in regard to finding community. Humans tend to gravitate toward other humans they have things in common with no matter what those "things" are, you know?

One of the more popular places for single men to stay in NYC was the YMCA, which had several residential buildings for young men. All the jokes you've probably heard about how gay men hung out at the Y and got up to their funny business there are based on fact. As rooming houses go, they were one of the safer options where legality was concerned as long as a man was discreet in his homosexuality and didn't draw undue attention to himself.

As opposed to a regular rooming house, the YMCA buildings offered largely short term rentals and most often to men who were new to the city. From Gay New York: Gender, Urban Culture, and the Making of the Gay Male World:

A man interviewed in the mid-1930s recalled of his stay at Sloane House [a YMCA residential building]:

"One night when I was coming in at 11:30 PM a stranger asked me to go to his room. They just live in one another's rooms although it's strictly forbidden... This YMCA is for transients but one further uptown [the West Side Y] is a more elegant brothel, for those who like to live in their ivory towers with Greek gods. If you go to a shower there is always someone waiting to have an affair. It doesn't take long."

And:

A man from New Jersey remembered that he stayed at Sloane House "many times, every chance I got... [because] it was very gay"; another man called it a "gay colony." Indeed the Y had such a reputation for sexual adventure that some New Yorkers took rooms at Sloane House for the weekend, giving fake out-of-town addresses.

So, in addition to being a place to find an affair for locals, YMCA residential buildings were also a gateway to the city's gay community because when a man got to a New York, it was often the first place he'd stay even if only temporarily. Of course, temporarily is all it takes to meet like-minded people and make a couple of contacts in the local community. From there he would meet other gay men and find his own place in that community.

When rooming houses started to go out of use, apartment buildings took up the space that they had occupied. An apartment where a man lived alone was a place he could take a fling home under the guise of 'just hanging out' or live with another man as 'roommates sharing expenses' without anyone being any the wiser. They were a big step up in that regard since rooming houses often offered little to no privacy. This added privacy also helped expand gay men's non-sexual social circle since in the comfort of your own home, you're more free both to have a leisurely hang out with friends and to talk about things you wouldn't want to talk about in shared spaces.

As for hook-ups, those were where a gay man was most in danger of getting himself into legal hot water, not least because the police knew popular 'cruising' spots and hotels/rooming houses that were willing to rent to two men for an hour or so and used that knowledge to entrap men who were looking for that type of relationship. While discretion was generally enough to stay out of trouble at the YMCA, entrapment was a constant worry for those who found their partners elsewhere.

Cruising the parks and such in search of someone (which New York made officially illegal in 1923) was the most common way to find those partners. An industry of rent-by-the-hour 'hotels' thrived along the edges of popular cruising spots like Battery Park and Times Square, among others - much to the displeasure of the city. They were willing to not ask questions and not call the police in order to make money (particularly in the 1930s, when making money wasn't so easy) Public baths were another place where a gay man could often find a short-term partner by making discreet inquiries.

Aside from bars that were frequented by gay men (often called 'pansy clubs' when they began to really make a splash in the 30s), cafeterias were a really popular place for gay men to spend social time together. Those who lived in rooming houses took most of their meals in public cafeterias and restaurants. Back to Gay New York for a long quote on the topic:

Particular restaurants served as the locus of particular gay social networks; overlapping groups of friends would meet regularly for dinner and camaraderie. The role of restaurants as social centers meant they often functioned as a crucial point of entry into the gay world for men just beginning to identify themselves as gay; for men already deeply involved in the gay world, they were a vital source of information about the gay scene, police activity, cultural events, and the like. The determination of gay men to claim space for themselves in the city's eating places - which they did boldly enough at some cafeterias to give them citywide reputations as "fairy hangouts," and surreptitiously enough at other places that they remained known only to other gay men - occasionally provoked a sharp reaction from social-purity forces. But gay men developed elaborate stratagems to protect such places, precisely because they played such an important role in their lives.

After WWII, ideas about gender and sexuality slowly started to change in the United States, and with less fear of the consequences of being found out (which were becoming more social than legal by the late 40s), men were able to find one another and form social groups (and support networks) more easily. This ability to organize gave rise to more open groups of gay men such as the Mattachine Society that was founded in 1950 and which served as both a place for gay men (and, to a lesser extent, lesbians) to get community information and a political force for gay rights.

So, that's the long answer to the question of how gay men (in NYC at least) avoided being arrested for being gay. The shorter answer is that they relied on discretion, painstakingly built community up around themselves, and looked out for one another.

Sources:

Chauncey, G. Gay New York: Gender, Urban Culture, and the Making of the Gay Male World
Bronski, M. A Queer History of the United States

And, so you don't have to find and read those:

This is an excerpt from Gay New York.
This talks about gay history in Brooklyn specifically.
This one is about gay culture in the 20s.
Here's something about the pansy club craze in Chicago and here's one about the same in LA, just to give a little more rounded view of the US.