Josephs was born in Green Street, South Shields, and all five brothers became referees. Arnold taught PE and Geography at what became the South Shields Boys High School from 1919-56. His refereeing career reached its peak in the '20s, when he was in charge of the Scotland v Ireland game at Celtic Park in 1926, and officiated at the FA Cup Final of 1929 (Bolton v Portsmouth). The same year he refereed the Scotland v England match at Hampden Park.
His career was cut short by injury, but he continued writing his column for the Daily Express entitled 'Knotty Soccer Problems'.