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I'll discuss models for distributed and private analysis of network data in which nodes retain control of their data. I'll focus particularly on the *local* node-private model, which is a good fit for distributed social network data. We provide the first formulation and systematic investigation of the model, design a new algorithmic framework tailored to it, and develop new lower bound techniques that show fundamental limitations on its power. Joint work with Sofya Raskhodnikova, Connor Wagaman, and Anatoly Zavyalov.