Existing reconstruction models in snapshot compressive imaging systems (SCI) are trained with a single well-calibrated hardware instance, making their performance vulnerable to hardware shifts and limited in adapting to multiple hardware configurations. To facilitate cross-hardware learning, previous efforts attempt to directly collect multi-hardware data and perform centralized training, which is impractical due to severe user data privacy concerns and hardware heterogeneity across different platforms/institutions. In this study, we explicitly consider data privacy and heterogeneity in cooperatively optimizing SCI systems by proposing a Federated Hardware-Prompt learning (FedHP) framework. Rather than mitigating the client drift by rectifying the gradients, which only takes effect on the learning manifold but fails to solve the heterogeneity rooted in the input data space, FedHP learns a hardware-conditioned prompter to align inconsistent data distribution across clients, serving as an indicator of the data inconsistency among different hardware (e.g., coded apertures). Extensive experimental results demonstrate that the proposed FedHP coordinates the pre-trained model to multiple hardware configurations, outperforming prevalent FL frameworks for 0.35dB under challenging heterogeneous settings. Moreover, a Snapshot Spectral Heterogeneous Dataset has been built upon multiple practical SCI systems. We will release the data and code.