Mitigating popularity bias is essential for two-sided fairness in recommendation systems. Existing methods usually require sensitive information such as demographic data and item attributes. However, due to policies and laws, such data are often inaccessible, limiting practical applications. To address this, we propose the Dual-Teacher Two-sided Fairness Distillation Framework (DTTFD), a teacher-student architecture that mitigates popularity bias without sensitive attributes. Specifically, we find that balancing user-item interactions or equalizing item exposures improves fairness, but inevitably sacrifices recommendation accuracy. Based on this insight, we utilize data augmentation to construct two unbiased datasets: one enforcing uniform user activities for user-side fairness, and the other ensuring equal item exposures for item-side fairness. We then train two fairness-specific teachers separately on these datasets and integrate them into a synthetic teacher, which distills fairness knowledge into the student model. Concurrently, an accuracy-focused teacher guides the student model to maintain high recommendation quality. Additionally, we introduce a momentum-based dynamic adjustment algorithm to adaptively balance fairness and accuracy throughout the distillation process. Experiments on three real-world datasets demonstrate significant fairness improvements over state-of-the-art methods while maintaining competitive recommendation accuracy.