Person re-identification (re-ID) is a fundamental task in intelligent surveillance and public safety. Federated learning (FL) provides a privacy-preserving paradigm for collaborative model training without centralized data collection. However, deploying FL in real-world re-ID systems remains challenging due to statistical heterogeneity caused by non-IID client data and the substantial communication overhead incurred by frequent transmission of large-scale models. To address these challenges, we propose FedKLPR, a lightweight and communication-efficient federated learning framework for person re-ID. FedKLPR consists of three key components. First, KL-Divergence-Guided training, including the KL-Divergence Regularization Loss (KLL) and KL-Divergence-aggregation Weight (KLAW), is introduced to mitigate statistical heterogeneity and improve convergence stability under non-IID settings. Second, unstructured pruning is incorporated to reduce communication overhead, and the Pruning-ratio-aggregation Weight (PRAW) is proposed to measure the relative importance of client parameters after pruning. Together with KLAW, PRAW forms KL-Divergence-Prune Weighted Aggregation (KLPWA), enabling effective aggregation of pruned local models under heterogeneous data distributions. Third, Cross-Round Recovery (CRR) adaptively controls pruning across communication rounds to prevent excessive compression and preserve model accuracy. Experiments on eight benchmark datasets demonstrate that FedKLPR achieves substantial communication savings while maintaining competitive accuracy. Compared with state-of-the-art methods, FedKLPR reduces communication cost by 40\%--42\% on ResNet-50 while achieving better overall performance.