We show that the largest density of factor of i.i.d. independent sets in the $d$-regular tree is asymptotically at most $(\log d)/d$ as $d\to\infty$. This matches the lower bound given by previous constructions. It follows that the largest independent sets given by local algorithms on random $d$-regular graphs have the same asymptotic density. In contrast, the density of the largest independent sets in these graphs is asymptotically $2(\log d)/d$. We prove analogous results for Poisson–Galton–Watson trees, which yield bounds for local algorithms on sparse Erdős–Rényi graphs.