The son of Stebbing Shaw (died 1799), rector of Hartshorne, Derbyshire, he was born near Stone, Staffordshire; his mother's maiden name was Hyatt, and she owned a small Staffordshire estate, which passed to her son.
Shaw was elected fellow of the Society of Antiquaries of London on 5 March 1795, and on 27 April 1799 he succeeded his father in the rectory of Hartshorne.