Production Studio
Paramount Pictures
The Short Version
Paramount Pictures distributed the movie, and Hitchcock shot the whole thing on their Hollywood soundstage.
The Slightly Less Short Version
Paramount is the second-oldest major Hollywood studio, arriving just after Universal Pictures in 1912. Back then, it was called Famous Players Film Company, formed by Adolph Zukor and the Frohman brothers. The pairing closely mirrored the film industry at the time: Zukor made nickelodeons, and the Frohmans were theatrical producers. This was back when the movies were far more of a novelty than an art form, and these guys' previous jobs show us the kind of rag-tag way film slowly evolved into art.
Paramount was their distribution partner, organized in 1914 as a means of getting the films out there. In 1916, they merged with the Jesse L. Lasky Feature Play Company to create the Famous Players-Lasky Corporation, and still used Paramount as their distribution arm. That famous studio logo with the mountain on it was in place in 1917, and it stuck.
Paramount never really had a genre specialty the way Warner Bros. made gangster pictures or Universal made monster movies. But it did believe in the star system, and some of the earliest screen idols—Rudolph Valentino, Douglas Fairbanks, and Mary Pickford, among others—all worked for the studio. That continued into the '30s and '40s, with the likes of Marlene Dietrich and the Marx Brothers taking over from their silent film predecessors. It even had cartoon characters like Popeye and Betty Boop to compete with that pesky mouse and his buddies over at Disney.
Its connection to theatrical distribution, in place since the studio's earliest days, actually got the company into trouble with the U.S. government. It more or less blackmailed theaters into taking a bunch of its cruddy, lesser pictures if they wanted to get the big movies featuring its stars. (In modern terms, imagine a theater not being able to show The Avengers unless it also showed Sharknado 6 throughout the run.)
The whole thing came to a head in a famous Supreme Court decision in 1948, United States vs. Paramount Pictures, which forbade movie studios from owning their own movie theaters. The court said it violated anti-trust laws designed to prevent monopolies. That hit Paramount hard, and after several decades of being a major player, it went into a slow decline. That was about where it was when Hitchcock was filming Rear Window.
Television ended up saving Paramount. It got in on the TV deal early, in 1939, and the 1960s saw its fortunes turn with the likes of Star Trek and Mission: Impossible (the TV show, not the movies). At the same time, a producer named Robert Evans was hired to head the studio, and he started to turn things around with blockbusters like Rosemary's Baby, The Odd Couple, Chinatown, and The Godfather, which was actually the all-time box office champ for a while until those pesky Lucas/Spielberg types showed up.
Paramount continued making hit films in all kinds of genres for many years, including comedies like Airplane!, action movies like Raiders of the Lost Ark, musicals like Footloose, gangster movies like The Untouchables, and all of those Star Trek movies still getting made today. Iron Man was theirs for a while until Disney snapped him back up. And, love 'em or hate 'em, the Transformers movies have made a ton of money for the folks on the mountain.
Hitchcock was a regular at Paramount in the 1950s, with the likes of Vertigo, To Catch a Thief, The Man Who Knew Too Much, and The Trouble With Harry coming out of the studio. The studio got to distribute Psycho, too, though Hitch shot it over at Universal because Paramount was spooked by the film's themes (which is why you can see Norman and his infamous motel when you take the Universal Studios Tour).