  We provide a detailed study of the estimation of probability
distributions---discrete and continuous---in a stringent setting in which data
is kept private even from the statistician. We give sharp minimax rates of
convergence for estimation in these locally private settings, exhibiting
fundamental tradeoffs between privacy and convergence rate, as well as
providing tools to allow movement along the privacy-statistical efficiency
continuum. One of the consequences of our results is that Warner's classical
work on randomized response is an optimal way to perform survey sampling while
maintaining privacy of the respondents.
