To protect privacy and meet legal regulations, federated learning (FL) has gained significant attention for training speech-to-text (S2T) systems, including automatic speech recognition (ASR) and speech translation (ST). However, the commonly used FL approach (i.e., \textsc{FedAvg}) in S2T tasks typically suffers from extensive communication overhead due to multi-round interactions based on the whole model and performance degradation caused by data heterogeneity among clients.To address these issues, we propose a personalized federated S2T framework that introduces \textsc{FedLoRA}, a lightweight LoRA module for client-side tuning and interaction with the server to minimize communication overhead, and \textsc{FedMem}, a global model equipped with a $k$-nearest-neighbor ($k$NN) classifier that captures client-specific distributional shifts to achieve personalization and overcome data heterogeneity. Extensive experiments based on Conformer and Whisper backbone models on CoVoST and GigaSpeech benchmarks show that our approach significantly reduces the communication overhead on all S2T tasks and effectively personalizes the global model to overcome data heterogeneity.