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Federated learning (FL) has become a key component in various language modeling applications such as machine translation, next-word prediction, and medical record analysis. These applications are trained on datasets from many FL participants that often include privacy-sensitive data, such as healthcare records, phone/credit card numbers, login credentials, etc. Although FL enables computation without necessitating clients to share their raw data, determining the extent of privacy leakage in federated language models is challenging and not straightforward. Moreover, existing attacks aim to extract data regardless of how sensitive or naive it is. To fill this research gap, we introduce two novel findings with regard to leaking privacy-sensitive user data from federated large language models. Firstly, we make a key observation that model snapshots from the intermediate rounds in FL can cause greater privacy leakage than the final trained model. Secondly, we identify that privacy leakage can be aggravated by tampering with a model's selective weights that are specifically responsible for memorizing the sensitive training data. We show how a malicious client can leak the privacy-sensitive data of some other users in FL even without any cooperation from the server. Our best-performing method improves the membership inference recall by 29% and achieves up to 71% private data reconstruction, evidently outperforming existing attacks with stronger assumptions of adversary capabilities., Comment: 20 pages (including bibliography and Appendix), Submitted to ACM CCS '24