The sampling problem under local differential privacy has recently been
studied with potential applications to generative models, but a fundamental
analysis of its privacy-utility trade-off (PUT) remains incomplete. In this
work, we define the fundamental PUT of private sampling in the minimax sense,
using the f-divergence between original and sampling distributions as the
utility measure. We characterize the exact PUT for both finite and continuous
data spaces under some mild conditions on the data distributions, and propose
sampling mechanisms that are universally optimal for all f-divergences. Our
numerical experiments demonstrate the superiority of our mechanisms over
baselines, in terms of theoretical utilities for finite data space and of
empirical utilities for continuous data space.