There is growing interest in unifying the streaming and full-context automatic speech recognition (ASR) networks into a single end-to-end ASR model to simplify the model training and deployment for both use cases. While in real-world ASR applications, the streaming ASR models typically operate under more storage and computational constraints - e.g., on embedded devices - than any server-side full-context models. Motivated by the recent progress in Omni-sparsity supernet training, where multiple subnetworks are jointly optimized in one single model, this work aims to jointly learn a compact sparse on-device streaming ASR model, and a large dense server non-streaming model, in a single supernet. Next, we present that, performing supernet training on both wav2vec 2.0 self-supervised learning and supervised ASR fine-tuning can not only substantially improve the large non-streaming model as shown in prior works, and also be able to improve the compact sparse streaming model.