  In this paper, we propose two new definitions of local differential privacy
for belief functions. One is based on Shafer's semantics of randomly coded
messages and the other from the perspective of imprecise probabilities. We show
that such basic properties as composition and post-processing also hold for our
new definitions. Moreover, we provide a hypothesis testing framework for these
definitions and study the effect of "don't know" in the trade-off between
privacy and utility in discrete distribution estimation.
