We introduce a novel class of non-stationary covariance functions for random fields on linear networks that allows both the variance and the correlation range of the random field to vary spatially. The proposed covariance functions are useful to model random fields with a spatial dependence that is locally isotropic with respect to the resistance metric, a distance that reflects the topology of the network. The framework admits explicit stochastic representations of the associated random fields and can be naturally extended to matrix-valued covariance functions for vector-valued random fields. We assess the statistical and computational performance of a weighted local likelihood estimator for the proposed models using synthetic data generated on the street network of the University of Chicago neighborhood.