Cross-domain sequential recommendation (CSR) has garnered significant attention. Current federated frameworks for CSR leverage information across multiple domains but often rely on user alignment, which increases communication costs and privacy risks. In this work, we propose FedCSR, a novel federated cross-domain sequential recommendation framework that eliminates the need for user alignment between platforms. FedCSR fully utilizes cross-domain knowledge to address the key challenges related to data heterogeneity both inter- and intra-platform. To tackle the heterogeneity of data patterns between platforms, we introduce Model Contrastive Learning (MCL) to reduce the gap between local and global models. Additionally, we design Sequence Contrastive Learning (SCL) to address the heterogeneity of user preferences across different domains within a platform by employing tailored sequence augmentation techniques. Extensive experiments conducted on multiple real-world datasets demonstrate that FedCSR achieves superior performance compared to existing baseline methods.