We consider a private discrete distribution estimation problem with one-bit communication constraint. The privacy constraints are imposed with respect to the local differential privacy and the maximal leakage. The estimation error is quantified by the worst-case mean squared error. We completely characterize the first-order asymptotics of this privacy-utility trade-off under the one-bit communication constraint for both types of privacy constraints by using ideas from local asymptotic normality and the resolution of a block design mechanism. These results demonstrate the optimal dependence of the privacy-utility trade-off under the one-bit communication constraint in terms of the parameters of the privacy constraint and the size of the alphabet of the discrete distribution.