Local learning, which trains a network through layer-wise local targets and losses, has been studied as an alternative to backpropagation (BP) in neural computation. However, its algorithms often become more complex or require additional hyperparameters because of the locality, making it challenging to identify desirable settings in which the algorithm progresses in a stable manner. To provide theoretical and quantitative insights, we introduce the maximal update parameterization ($μ$P) in the infinite-width limit for two representative designs of local targets: predictive coding (PC) and target propagation (TP). We verified that $μ$P enables hyperparameter transfer across models of different widths. Furthermore, our analysis revealed unique and intriguing properties of $μ$P that are not present in conventional BP. By analyzing deep linear networks, we found that PC's gradients interpolate between first-order and Gauss-Newton-like gradients, depending on the parameterization. We demonstrate that, in specific standard settings, PC in the infinite-width limit behaves more similarly to the first-order gradient. For TP, even with the standard scaling of the last layer, which differs from classical $μ$P, its local loss optimization favors the feature learning regime over the kernel regime.