Local differential privacy (LDP) is increasingly employed in
privacy-preserving machine learning to protect user data before sharing it with
an untrusted aggregator. Most LDP methods assume that users possess only a
single data record, which is a significant limitation since users often gather
extensive datasets (e.g., images, text, time-series data) and frequently have
access to public datasets. To address this limitation, we propose a locally
private sampling framework that leverages both the private and public datasets
of each user. Specifically, we assume each user has two distributions: $p$ and
$q$ that represent their private dataset and the public dataset, respectively.
The objective is to design a mechanism that generates a private sample
approximating $p$ while simultaneously preserving $q$. We frame this objective
as a minimax optimization problem using $f$-divergence as the utility measure.
We fully characterize the minimax optimal mechanisms for general
$f$-divergences provided that $p$ and $q$ are discrete distributions.
Remarkably, we demonstrate that this optimal mechanism is universal across all
$f$-divergences. Experiments validate the effectiveness of our minimax optimal
sampler compared to the state-of-the-art locally private sampler.