Drought has become a critical global threat with significant societal impact. Existing drought monitoring solutions primarily focus on assessing drought severity using quantitative measurements, overlooking the diverse societal impact of drought from human-centric perspectives. Motivated by the collective intelligence on social media and the computational power of AI, this paper studies a novel problem of socially informed AI-driven drought estimation that aims to leverage social and news media information to jointly estimate drought severity and its societal impact. Two technical challenges exist: 1) How to model the implicit temporal dynamics of drought societal impact. 2) How to capture the social-physical interdependence between the physical drought condition and its societal impact. To address these challenges, we develop SIDE, a socially informed AI-driven drought estimation framework that explicitly quantifies the societal impact of drought and effectively models the social-physical interdependency for joint severity-impact estimation. Experiments on real-world datasets from California and Texas demonstrate SIDE's superior performance compared to state-of-the-art baselines in accurately estimating drought severity and its societal impact. SIDE offers valuable insights for developing human-centric drought mitigation strategies to foster sustainable and resilient communities.