They seem to have accomplished this almost simultaneously. Thompson published his first suspense novel in 1949; Highsmith published hers in 1950. Thompson published arguably his greatest work, The Killer Inside Me, in 1952; Highsmith published hers, The Talented Mr. Ripley, in 1955.
Their careers ran parallel in several ways. He published twenty-five novels; she published 22. Both of them saw many adaptations for film and television, and their stories continue to be adapted to this day, although Highsmith had notably better luck in her lifetime, beginning with her first novel, Strangers on a Train, which was the basis for Alfred Hitchcock's classic film of the same title.
Highsmith was born in Fort Worth, Texas, in 1921 and grew up there. At that time, Thompson, lived in Fort Worth, working odd jobs, selling his first stories, and gathering experiences that would turn up in his novels.
But for all that, they seem to have been unaware of each other. Andrew Wilson's biography of Highsmith, Beautiful Shadows, makes no mention of Thompson; Michael J. MacCauley's biography of Thompson, Sleep with the Devil, makes no mention of Highsmith.
Their careers also differ in several ways. Highsmith methodically published a novel about every other year throughout her adult life. Thompson published almost half his novels in just three years: two in 1952, five in 1953, and five in 1954. His biographer, MacCauley, notes these were essentially the only years of his life when he was sober.
The most important difference between them was the point of view they used in their novels. Highsmith wrote in the third person, thus referring to the hero-bad-guy as "he," and she had a good reason for doing so: "I have quite a bit of introspection in my heroes, and to write all this in the first person makes them sound like nasty schemers, which of course they are, but they seem less so if some all-knowing author is telling what is going on in their heads."
Apparently Thompson wasn't concerned what his bad-guys sounded like. He wrote in the first person, referring to the hero as "I." Here's a bit of introspection from Nick Corey the hero of Pop. 1280, recalling why his father beat him, "The fact was, I guess, that he just couldn't stand for me to be any good. If I was any good, then I couldn't be the low-down monster that had killed my own mother in getting born. And I had to be that. He had to have someone to blame."