We study parametric change-point detection under local differential privacy, focusing on the offline localization of a single change point in a sequence of independent observations.~In the non-private setting, we derive improved finite-sample accuracy guarantees for a generalized likelihood-ratio estimator using martingale methods.~We establish finite-sample minimax lower bounds in both the non-private and locally private settings, showing that localization accuracy is fundamentally limited by the Chernoff information between the pre- and post-change distributions.~Guided by this, we design locally private CPD algorithms based on aggregated randomized response and binary mechanisms, which maximize the Chernoff information between the induced output distributions in low- and high-privacy regimes, respectively, and derive their finite-sample guarantees. Together, our upper and lower bounds quantify the statistical cost of privacy in change-point localization.~As part of our analysis, we also show that strong data processing inequality coefficients of Rényi divergence and Jeffreys--Rényi divergence of order infinity can be computed by restricting to pairs of input distributions with a common support of at most two symbols.~We corroborate our results through empirical evaluation on synthetic data.

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