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. Copyright © 2022, The Authors. All rights reserved.