We analyze two variants of Local Gradient Descent applied to distributed logistic regression with heterogeneous, separable data and show convergence at the rate $O(1/KR)$ for $K$ local steps and sufficiently large $R$ communication rounds. In contrast, all existing convergence guarantees for Local GD applied to any problem are at least $Ω(1/R)$, meaning they fail to show the benefit of local updates. The key to our improved guarantee is showing progress on the logistic regression objective when using a large stepsize $η\gg 1/K$, whereas prior analysis depends on $η\leq 1/K$.