Baldwin shot Halyna Hutchins, the photography director of "Rust", and Joel Souza, the film's director, at the Bonanza Creek Ranch, a production location south of Santa Fe, according to the Santa Fe Sheriff's Department.
Hutchins was transported by helicopter to the University of New Mexico Hospital, where she was pronounced dead.
Souza was taken by ambulance to Christus St. Vincent Regional Medical center to undergo treatment for his injuries.
The severity of his injuries could not immediately be determined.
Actress Frances Fisher said on Twitter that "Souza texted me that he's out of hospital." Asked whether Souza had been discharged, medical center spokesperson Arturo Delgado said he was not allowed to release information about patients.
The Sheriff's office said that no charges have been filed and they are investigating the shooting and interviewing witnesses.
"The investigation remains open and active," the Sherrif's office said in a statement.
Entertainment news site Deadline cited a source in the Sheriff's office as saying that Baldwin was questioned by investigators and later released.
Baldwin went to the sheriff's office willingly and provided a statement to investigators, the Santa Fe New Mexican reported, citing spokesperson Juan Rios.
Deputies were still trying to determine whether what happened was an accident, the newspaper added. Rios did not immediately respond to requests for information from Reuters.
Baldwin's representatives did not immediately respond to requests for comment. Earlier on Thursday, news magazine People reported that a spokesperson for Baldwin had said there was an "accident" involving the "misfire of a prop gun with blanks."
Baldwin was seen "distraught and in tears" as he spoke on the phone outside the sheriff's office headquarters on Thursday, the Santa Fe New Mexican wrote.
Baldwin, 63, is a co-producer of "Rust", a Western movie set in 1880s Kansas, and plays the eponymous character who is an outlaw grandfather of a 13-year-old boy convicted of an accidental killing.
Production of the film had been halted for an "undetermined period," several news outlets quoted the film's production company, Rust Movie Productions LLC, as saying.
An e-mail to an address for the film production, which was listed on a New Mexico government statement, went unanswered.
Earlier on Thursday, Baldwin posted a picture of himself on Instagram sporting a grey beard and dressed in Western cowboy-style attire in front of trailers. He appeared to have a fake blood stain on his shirt and jacket.
"Back to in person at the office. Blimey ... it's exhausting," he wrote. The post was deleted late on Thursday night.
Known for his impersonations of former U.S. President Donald Trump on NBC's comedy sketch show "Saturday Night Live," Baldwin has a long history in film and television, including roles in "Glengarry Glen Ross" and "30 Rock."
Baldwin was charged in 2018 after a fight over a New York parking spot. He pleaded guilty to second-degree harassment and agreed to participate in an anger management program.
In 2014, he was given a summons for disorderly conduct after an argument with police who stopped him from riding his bike down a one-way street in New York. And in 2011, he was thrown off a plane for refusing to stop playing the game "Words with Friends" before take-off.
Hutchins, 42, who was originally from Ukraine and grew up on a Soviet military base in the Arctic Circle, once worked as an investigative reporter in Europe, according to her website.
She graduated from the American Film Institute in 2015 and was selected as one of American Cinematographer's Rising Stars of 2019, according to her website biography.
She described herself as a "Restless Dreamer" and an "Adrenaline junkie" on her Instagram page.
Her last post, two days ago, shows her grinning under a wide-brimmed hat as she rides a horse. "One of the perks of shooting a western is you get to ride horses on your day off:)" she captioned the video.
April Wright, a writer, director and producer, paid tribute to her on Facebook.
"I'm in disbelief," wrote Wright. "So young, vibrant, and talented. Such a wonderful soul. My heart goes out to her son and family."
Representatives for Hutchins did not immediately respond to a request for information about her death.
Souza, 48, lives in the San Francisco Bay Area with his wife and two children, according to the IMDB website. His LinkedIn page credits him as the "writer/director" of action film Crown Vic and comedy Christmas Trade.
"This is still an active investigation and we do not yet have all the facts," said SAG-AFTRA, which describes itself as the world's biggest labor union representing performers and broadcasters.
"We will continue to work with production, the other unions, and the authorities to investigate this incident and to understand how to prevent such a thing from happening again."
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