The celebrated palette sparsification result of [Assadi, Chen, and Khanna SODA'19] shows that to compute a $Δ+1$ coloring of the graph, where $Δ$ denotes the maximum degree, it suffices if each node limits its color choice to $O(\log n)$ independently sampled colors in $\{1, 2, \dots, Δ+1\}$. They showed that it is possible to color the resulting sparsified graph -- the spanning subgraph with edges between neighbors that sampled a common color, which are only $\tilde{O}(n)$ edges -- and obtain a $Δ+1$ coloring for the original graph. However, to compute the actual coloring, that information must be gathered at a single location for centralized processing. We seek instead a local algorithm to compute such a coloring in the sparsified graph. The question is if this can be achieved in $\operatorname{poly}(\log n)$ distributed rounds with small messages. Our main result is an algorithm that computes a $Δ+1$-coloring after palette sparsification with $O(\log^2 n)$ random colors per node and runs in $O(\log^2 Δ+ \log^3 \log n)$ rounds on the sparsified graph, using $O(\log n)$-bit messages. We show that this is close to the best possible: any distributed $Δ+1$-coloring algorithm that runs in the LOCAL model on the sparsified graph, given by palette sparsification, for any $\operatorname{poly}(\log n)$ colors per node, requires $Ω(\log Δ/ \log\log n)$ rounds. This distributed palette sparsification result leads to the first $\operatorname{poly}(\log n)$-round algorithms for $Δ+1$-coloring in two previously studied distributed models: the Node Capacitated Clique, and the cluster graph model.