Multiple players are each given one independent sample, about which they can\nonly provide limited information to a central referee. Each player is allowed\nto describe its observed sample to the referee using a channel from a family of\nchannels $\\mathcal{W}$, which can be instantiated to capture both the\ncommunication- and privacy-constrained settings and beyond. The referee uses\nthe messages from players to solve an inference problem for the unknown\ndistribution that generated the samples. We derive lower bounds for sample\ncomplexity of learning and testing discrete distributions in this\ninformation-constrained setting.\n Underlying our bounds is a characterization of the contraction in chi-square\ndistances between the observed distributions of the samples when information\nconstraints are placed. This contraction is captured in a local neighborhood in\nterms of chi-square and decoupled chi-square fluctuations of a given channel,\ntwo quantities we introduce. The former captures the average distance between\ndistributions of channel output for two product distributions on the input, and\nthe latter for a product distribution and a mixture of product distribution on\nthe input. Our bounds are tight for both public- and private-coin protocols.\nInterestingly, the sample complexity of testing is order-wise higher when\nrestricted to private-coin protocols.\n