We study the behavior of the taxicab and Euclidean metrics on permutations with a prescribed peak set. Given a set \( S \subseteq \{1, \ldots , n\} \), we compute the diameter of the set \( P (S; n) \) of permutations having peak set \( S \) with respect to these metrics. For the taxicab metric, we determine the pairs of permutations attaining the diameter and provide a characterization of those peak sets \( S \) for which the diameter of \( P (S; n) \) is maximal or minimal, in terms of suitable partitions of \( [n] \). For the Euclidean metric, the diameter of \( \mathfrak S_n \) is attained by complementary permutations. We analyze how the peak set restriction affects this phenomenon and show that the diameter of \( P (S; n) \) depends on both the number of peaks and their relative positions. As a consequence, we obtain an explicit formula for the Euclidean diameter of \( P (S; n) \). Finally, we show that these results admit natural extensions to signed permutations of type \( B \), yielding explicit formulas for the diameters of the corresponding peak classes.