We present SODA: the first publicly available, million-scale high-quality social dialogue dataset. Using SODA, we train COSMO: a generalizable conversation agent outperforming previous best-performing agents on both in- and out-of-domain datasets. In contrast to most existing crowdsourced, small-scale dialogue corpora, we distill 1.5M socially-grounded dialogues from a pre-trained language model (InstructGPT; Ouyang et al., 2022). Dialogues are distilled by contextualizing social commonsense knowledge from a knowledge graph (Atomic10x; West et al., 2022). Human evaluation shows that dialogues in SODA are more consistent, specific, and (surprisingly) natural than prior human-authored datasets - e.g., DailyDialog (Li et al., 2017), BlendedSkillTalk (Smith et al., 2020). In addition, extensive evaluations show that COSMO is significantly more natural and consistent on unseen datasets than best-performing dialogue models - e.g., GODEL (Peng et al., 2022), BlenderBot (Roller et al., 2021), DialoGPT (Zhang et al., 2020). Furthermore, it is sometimes even preferred to the original human-written gold responses. We make our data, models, and code public.