Dataset Distillation (DD) is a powerful technique for reducing large datasets into compact, representative synthetic datasets, accelerating Machine Learning training. However, traditional DD methods operate in a centralized manner, which poses significant privacy threats and reduces its applicability. To mitigate these risks, we propose a Secure Federated Data Distillation (SFDD) framework to decentralize the distillation process while preserving privacy. Unlike existing Federated Distillation techniques that focus on training global models with distilled knowledge, our approach aims to produce a distilled dataset without exposing local contributions. We leverage the gradient-matching-based distillation method, adapting it for a distributed setting where clients contribute to the distillation process without sharing raw data. The central aggregator iteratively refines a synthetic dataset by integrating client-side updates while ensuring data confidentiality. To make our approach resilient to inference attacks perpetrated by the server that could exploit gradient updates to reconstruct private data, we create an optimized Local Differential Privacy approach, called LDPO-RLD. Furthermore, we assess the framework's resilience against malicious clients executing backdoor attacks (such as Doorping) and demonstrate robustness under the assumption of a sufficient number of participating clients. Our experimental results demonstrate the effectiveness of SFDD and that the proposed defense concretely mitigates the identified vulnerabilities, with minimal impact on the performance of the distilled dataset. By addressing the interplay between privacy and federation in dataset distillation, this work advances the field of privacy-preserving Machine Learning making our SFDD framework a viable solution for sensitive data-sharing applications.