Hearst is one of the five daughters of Randolph Apperson Hearst (1915–2000), former president of The San Francisco Examiner, and his first wife, the former Catherine Wood Campbell. She is the granddaughter of newspaper tycoon William Randolph Hearst. Patricia Hearst, who was kidnapped in 1974 by members of the Symbionese Liberation Army, is one of her sisters.

In March 1975, Hearst was arrested and charged with misdemeanor possession of crystallized amphetamine. She was arrested at Niagara Falls, NY, along with Daniel Moffet, one of two other passengers in the car she was driving when the trio entered the United States at the Rainbow bridge. Both of them were later released on $1,000 recognizance bond. During the court hearing, both Moffet and Hearst said that the pills were hers, and so the US attorney’s office recommended the charges against Moffet be dropped. Hearst was also questioned by FBI agents about her sister Patty Hearst, a then fugitive. Although there was speculation that Anne had visited her sister Patricia while in Canada, authorities said she was merely driving from Detroit to New York City and the Canadian route was the shortest. After serving four months’ probation, the charges against her were dropped, as she had properly served her time of probation without violating any of its conditions.