The advent of smart meters has enabled pervasive collection of energy consumption data for training short-term load forecasting models. In response to privacy concerns, federated learning (FL) has been proposed as a privacy-preserving approach for training, but the quality of trained models degrades as client data becomes heterogeneous. In this paper we propose the use of personalization layers for load forecasting in a general framework called PL-FL. We show that PL-FL outperforms FL and purely local training, while requiring lower communication bandwidth than FL. This is done through extensive simulations on three different datasets from the NREL ComStock repository.