We study estimation of a deterministic binary count from the shuffled outputs of a common single-message local channel. Privacy is required for every neighboring composition at fixed global parameters $(\varepsilon,δ)$, and estimators may be biased. We obtain uniform critical Poisson and Skellam approximations and characterize the limiting minimax risk as an infimum over finite observed-mark Poisson experiments. Let $Λ$ be the noise intensity required by the boundary Poisson-shift profile and let $λ_{\min}$ be the intensity required by the full composition-maximized randomized-response profile. Binary randomized response is asymptotically minimax optimal if and only if $Λ=λ_{\min}$. When $Λ<λ_{\min}$, the minimax value lies strictly between $(Λ+λ_{\min})/2$ and $λ_{\min}$, and a randomized-response construction with observed rare marks strictly improves on binary randomized response. We give a finite-sample certified example. Binary randomized response is strictly suboptimal at $\varepsilon=0$ for every $0<δ<1$, and this persists for sufficiently small positive $\varepsilon$ at each fixed target. For every fixed positive $\varepsilon$, strict suboptimality also occurs along a sequence of fixed privacy targets tending to zero. The variational characterization permits approximation by finite marked channels; it does not assert that a finite optimizer exists.

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