There was a slightly messy bridge period between "gay" in the sense of bright or happy and "gay" in the sense of homosexual. In the 1880s and 1890s, "gay" had acquired connotations of flashiness or tawdriness at least in certain underground contexts -- one could still describe innocent pastimes as gay, but women of diminished virtue could be said to favor gay clothing, and prostitutes' workplaces might be described as gay houses. This is the product of 19th century understandings of why women engaged in prostitution, as well as an illustration of those understandings -- women with poor morals were light-minded and inconstant, they wanted to wear flashy bright clothes that were incongruous with their social standing, and to get those clothes they engaged in prostitution. Some of this spilled over to encompass male prostitutes, who might dress in female trappings or might not. The criminal edge of this sense of "gay" made its way into 1890s hobo slang in the form of "gaycat" -- a less-experienced hobo traveling with a more-experienced hobo, with the connotation of a homosexual relationship between these two. (In this way it parallels the migration of "punk" -- first a term to describe female prostitutes and loose women, later a term used to describe an experienced criminal's male apprentice with a similar sexual connotation and then later after that used to describe simply an inexperienced young person or a young criminal.)

By the first half of the 20th century "gay" was still entrenched in its mainstream meaning, but for those in the know its double meaning was a way to gauge another person's familiarity with homosexuality and homosexual subcultures. Thomas Painter in an unpublished 1941 manuscript ("Male Homosexuals and their Prostitutes in Contemporary America") describes the use of double entendre:

Supposing one met a stranger on a train from Boston to New York and wanted to find out whether he was "wise" or even homosexual. One might ask: "Are there any gay spots in Boston?" And by slight accent put on the word "gay" the stranger, if wise, would understand that homosexual resorts were meant. The uninitiated stranger would never suspect, inasmuch as "gay" is also a perfectly normal and natural word to apply to places where one has a good time.

(1941 was the same year Warner Bros. released The Gay Parisian, by the way, and RKO put out its first film about suave gentleman detective The Gay Falcon.) By the 1940s, men might self-identify as "gay", but it was one of a number of terms in circulation for the first few decades, each with a slightly different shade of meaning -- "pansy", "fairy", "queer" as both an adjective and a noun, etc. In official documents, these same men might be identified as "[sexual] perverts", "inverts, "degenerates", or "homosexualists", more or less interchangeably. The use of "gay" still had a tinge of its older meaning when used to describe places where homosexual men and women might congregate -- these places were colorful and festive, "places where one has a good time", but they were also somewhat cheap and fast. But above all it seems to have been in use because it was firmly unpretentious; while it had connotations of a seamy urban underworld, it didn't have the immediate and arresting antisocial connotations of, say, describing oneself as a degenerate or one's friends as perverts. The function of double entendre "gay" had with the mainstream sense of the word was an asset, but only to those who were already in the know, already in the gay life.

When people want a cinematic milestone for the use of "gay" to mean gay, they often cite Cary Grant in the 1938 film Bringing Up Baby. Brandie Ashe's blog post puts it better than I could:

In 1938’s Bringing Up Baby, when Grant’s character, David, accompanies Susan (Katharine Hepburn) and her leopard, Baby, to Connecticut, she convinces him to take a shower in order to delay his return to New York. While he showers, she steals his clothes and sends them into town to be cleaned. In dismay, David throws on Susan’s frilly, feather-trimmed robe and runs into Susan’s aunt (May Robson). Aunt Elizabeth, shocked to find a negligee-wearing man in her niece’s house, demands to know why he’s wearing women’s clothing, and an increasingly frustrated David finally leaps into the air, shouting, “Because I just went GAY all of a sudden!”

Here the mainstream sense of "gay" is essential to the joke getting into theaters -- "I just decided to do something fun and frivolous!" has more plausible deniability than "because I felt like being a fairy!" -- but it's enough to seal this moment's place in works like Vito Russo's The Celluloid Closet, helped along by historical speculation that both Hepburn and Grant were bisexual. Other men as well as women identified as "gay" in psychological documentation and in criminal proceedings, but the sense still drew from the idea of being in "gay circles" and attending "gay nightspots", rather than all homosexuals being gay as a matter of course and the terms being direct synonyms. Etymonline cites from 1947:

After discharge A.Z. lived for some time at home. He was not happy at the farm and went to a Western city where he associated with a homosexual crowd, being "gay," and wearing female clothes and makeup. He always wished others would make advances to him. ["Rorschach Research Exchange and Journal of Projective Techniques," 1947, p.240]

By 1947, the wonderful Edith Eyde gave her lesbian periodical Vice Versa the teasing subtitle "America's Gayest Magazine" -- clearly playing on both senses. The magazine is a place for lesbians, women who love women and who might attend gay bars, to let their hair down and have a little fun as well as discuss serious topics. (The imagery of "letting one's hair down" has a rich history in homosexual slang from this period, but that's besides the point.) By the early 1950s, risque comedic musician Ruth Wallis could play on the word's two senses in songs like "De Gay Young Lad From Trinidad" -- leveraging the double entendre between mainstream music's innocuous use of "gay" to mean "happy and carefree" and the fact that she herself obviously did not. (Ditto with other slang -- the use of "fairy", "pansy", "queen" and "queer" in her song "Queer Things" relies on a similar familiarity with both senses for humor.) The innocence of the mainstream term -- having largely shed its connotations of fastness and sordidness -- just made the double sense funnier, to people who were predisposed to find that kind of thing funny.

So when did "gay" stop meaning gay and start meaning, well, gay? When did straight people quit using gay in the sense of this 1946 Jester Wools ad for fear of running into the double entendre? In the late-1960s and 1970s, when gay liberation activism began to intrude into mainstream heterosexual consciousness. You might compare this to the shift in the dominant connotation of "queer" -- the shift to "queer" in activist language was a product of internal shifts in LGBT activism and dialogue in the 1980s; "queer" had been used as a pejorative for gay people long before the 1980s, but its indelible association with homosexual activism and communities rather than chance use to describe the strangeness and bizarreness of homosexuals was the last nail in the coffin of its innocuous use as neither a slur nor a self-identifier.

George Chauncey's Gay New York gives a rough overview of the shifting gay nomenclature of the first half of the 20th century and shades of meaning within those terms; Henry L. Minton's chapter on Thomas Painter in Departing From Deviance: A History of Homosexual Rights And Emancipatory Science in America gives a sense of what Minton's manuscript was trying to document and why.