Medical imaging provides many valuable clues involving anatomical structure and pathological characteristics. However, image degradation is a common issue in clinical practice, which can adversely impact the observation and diagnosis by physicians and algorithms. Although extensive enhancement models have been developed, these models require a well pre-training before deployment, while failing to take advantage of the potential value of inference data after deployment. In this paper, we raise an algorithm for source-free unsupervised domain adaptive medical image enhancement (SAME), which adapts and optimizes enhancement models using test data in the inference phase. A structure-preserving enhancement network is first constructed to learn a robust source model from synthesized training data. Then a teacher-student model is initialized with the source model and conducts source-free unsupervised domain adaptation (SFUDA) by knowledge distillation with the test data. Additionally, a pseudo-label picker is developed to boost the knowledge distillation of enhancement tasks. Experiments were implemented on ten datasets from three medical image modalities to validate the advantage of the proposed algorithm, and setting analysis and ablation studies were also carried out to interpret the effectiveness of SAME. The remarkable enhancement performance and benefits for downstream tasks demonstrate the potential and generalizability of SAME. The code is available at https://github.com/liamheng/Annotation-free-Medical-Image-Enhancement.