Sequential recommender systems have made significant progress. Recently, due to increasing concerns about user data privacy, some researchers have implemented federated learning for sequential recommendation, a.k.a., Federated Sequential Recommender Systems (FedSeqRecs), in which a public sequential recommender model is shared and frequently transmitted between a central server and clients to achieve collaborative learning. Although these solutions mitigate user privacy to some extent, they present two significant limitations that affect their practical usability: (1) They require a globally shared sequential recommendation model. However, in real-world scenarios, the recommendation model constitutes a critical intellectual property for platform and service providers. Therefore, service providers may be reluctant to disclose their meticulously developed models. (2) The communication costs are high as they correlate with the number of model parameters. This becomes particularly problematic as the current FedSeqRec will be inapplicable when sequential recommendation marches into a large language model era. To overcome the above challenges, this paper proposes a parameter transmission-free federated sequential recommendation framework (PTF-FSR), which ensures both model and data privacy protection to meet the privacy needs of service providers and system users alike. Furthermore, since PTF-FSR only transmits prediction results under privacy protection, which are independent of model sizes, this new federated learning architecture can accommodate more complex and larger sequential recommendation models. Extensive experiments conducted on three widely used recommendation datasets, employing various sequential recommendation models from both ID-based and ID-free paradigms, demonstrate the effectiveness and generalization capability of our proposed framework.