In November 2020, Twitter co-founder Jack Dorsey picked a famed ex-hacker, Peiter Zatko, to solve some of his social-media company’s most pernicious problems: protecting user privacy and the security of its computer systems.

His choice was a man with near-legendary status in the industry who had been finding flaws and ways to correct them for nearly 30 years. Mr. Zatko, known as “Mudge,” is now alleging widespread dysfunction and flaws in the department he oversaw and was hired to fix.