Advanced by rich perception and precise execution, robots possess immense potential to provide professional and customized rehabilitation exercises for patients with mobility impairments caused by strokes. Autonomous robotic rehabilitation significantly reduces human workloads in the long and tedious rehabilitation process. However, training a rehabilitation robot is challenging due to the data scarcity issue. This challenge arises from privacy concerns (e.g., the risk of leaking private disease and identity information of patients) during clinical data access and usage. Data from various patients and hospitals cannot be shared for adequate robot training, further compromising rehabilitation safety and limiting implementation scopes. To address this challenge, this work developed a novel federated joint learning (FJL) method to jointly train robots across hospitals. FJL also adopted a long short-term memory network (LSTM)-Transformer learning mechanism to effectively explore the complex tempo-spatial relations among patient mobility conditions and robotic rehabilitation motions. To validate FJL's effectiveness in training a robot network, a clinic-simulation combined experiment was designed. Real rehabilitation exercise data from 200 patients with stroke diseases (upper limb hemiplegia, Parkinson's syndrome, and back pain syndrome) were adopted. Inversely driven by clinical data, 300,000 robotic rehabilitation guidances were simulated. FJL proved to be effective in joint rehabilitation learning, performing 20% - 30% better than baseline methods.