This isn't my area of expertise, but I'll do my best to answer your question from the perspective of New York City history. After immigrating to America, the Capone family settled in Brooklyn (at 95 Navy Avenue), moving to Park Slope in 1910 (21 Garfield Avenue). After leaving school, Capone joined a series of colorfully named gangs: the Navy Street Boys, the James Street gang, the South Brooklyn Rippers, the Forty Thieves and the Five Points Juniors. The head of the Forty Thieves, Frankie Yale, ran a club called the Harvard Inn on Coney Island, and he installed Capone there as a bartender. One night in 1917, Capone told a young woman that he found her derriere beautiful, and her older brother took his knife and sliced Capone's left cheek, earning him the Scarface moniker.

After this altercation, Capone worked as a loan shark in the city under the auspices of Yale. In 1919, he got into a violent altercation with Criss Cross Finnegan and was targeted by Wild Bill Lovett, head of the White Hand gang, who only knew to looked for a "scarred man." Capone went into hiding, and Yale contacted his old boss, Johnny Torrio, who had went to Chicago to help out his uncle, Big Jim Colosimo. I'll let someone with more expertise in Chicago or crime history take it from there with a more detailed history, but he was definitely relatively well known and infamous in Brooklyn.

To answer your question directly, the average person (especially anyone who visited a speakeasy or tea pad) probably knew at least a little bit about the world of organized crime, as they were participating in it (one of my favorite facts is that Harlem had more than 500 known speakeasies in the 1920s). The newspapers clamored to report on the salacious details of gang violence and murders, and followed developments closely, as the mobsters murdered each other in increasing numbers during the Chicago "beer wars." New Yorkers loved their newspapers, bought from corner Newsboys and read on the subways. The Times reported on Capone's Chicago exploits, including reports from 1926 linking murders in Chicago to gunmen from New York's Cherry Hill gang, observing that Scarface Al "Caponi" escaped to Canada and reporting on Capone showing up for a disorderly conduct charge, "immaculately groomed, even to spats and a cane" and surrounded by surly bodyguards. Tabloids and penny newspapers repeated such stories (often getting them from the Chicago dailies) to sell more newspapers. It is likely that a fair amount of New Yorkers knew about Capone's exploits well before he was taken down by the Feds.