Modern AI models have shown remarkable potential for enhancing diagnostic accuracy through automated medical image analysis across diverse clinical tasks. However, many healthcare institutions lack sufficient data, computational infrastructure, or technical expertise to develop such models independently. As a result, they must rely on externally developed models, which are often proprietary and restricted by commercial, regulatory, or institutional constraints. Even when accessible, direct deployment of these models typically leads to degraded performance due to distributional shifts between source and local data. To address these limitations, we propose CARE (Calibration-Aware REliable collaboration), an output-level collaboration framework that enables under-resourced institutions to make use of the predictive outputs from external AI models without requiring access to their internal parameters. In CARE, source institutions provide black-box logit predictions via APIs, ensuring that both model parameters and architectures remain private, while enabling seamless reuse of pretrained models by target institutions. At the target institution, a lightweight calibrator is trained on a small, locally labeled dataset to adapt predictions to the local data distribution and assign sample-specific uncertainty scores. The calibrated outputs are then fused using Dempster-Shafer theory, facilitating fine-grained reliability control beyond static model-level weighting. Notably, CARE supports bidirectional collaboration: target institutions obtain locally adapted and reliable predictions with minimal computational overhead, while source institutions receive aggregated feedback from real-world deployment, fostering continual model improvement and generalizability. Experiments on medical image classification tasks demonstrate that CARE achieves up to 12.24% accuracy gain over state-of-the-art collaboration methods under substantial domain shifts, highlighting its effectiveness in addressing real-world heterogeneity in clinical AI deployment. The source code is publicly available at https://github.com/KylinYang70/CARE .