Network slicing is one of the cornerstones for next-generation mobile communication systems. Specifically, it enables Mobile Virtual Network Operators (MVNOs) to offer various types of services over the same physical infrastructure owned by an Infrastructure Provider (InP). To satisfy the dynamic user requirements and ensure resource efficiency, MVNOs need to estimate the future traffic demand in advance, to pre-allocate/reconfigure the resources at the base stations. However, this per-slice traffic forecasting exploits information that is clearly sensitive for the MVNOs from a business point of view, and which might even disclose private data regarding some users. Hence, it is vital for MVNOs to ensure data privacy while conducting traffic forecasting. Bearing this in mind, we propose the Federated Proximal Long Short-Term Memory (FPLSTM) framework, which allows MVNOs to train their local models with their private dataset at each base station without compromising data privacy. Simultaneously, an InP global model is updated through the aggregation of local models weights. Prediction results obtained by training the models on a real-world dataset indicate that the forecasting performance of FPLSTM is as accurate as state-of-the-art solutions, while ensuring data privacy, computation and communication cost efficiency.