AI creates and exacerbates privacy risks, yet practitioners lack effective resources to identify and mitigate these risks. We present Privy, a tool that guides practitioners without privacy expertise through structured privacy impact assessments to: (i) identify relevant risks in novel AI product concepts, and (ii) propose appropriate mitigations. Privy was shaped by a formative study with 11 practitioners, which informed two versions -- one LLM-powered, the other template-based. We evaluated these two versions of Privy through a between-subjects, controlled study with 24 separate practitioners, whose assessments were reviewed by 13 independent privacy experts. Results show that Privy helps practitioners produce privacy assessments that experts deemed high quality: practitioners identified relevant risks and proposed appropriate mitigation strategies. These effects were augmented in the LLM-powered version. Practitioners themselves rated Privy as being useful and usable, and their feedback illustrates how it helps overcome long-standing awareness, motivation, and ability barriers in privacy work.