Medical image denoising faces great challenges. Although deep learning methods have shown great potential, their efficiency is severely affected by millions of trainable parameters. The non-linearity of neural networks also makes them difficult to be understood. Therefore, existing deep learning methods have been sparingly applied to clinical tasks. To this end, we integrate known filtering operators into deep learning and propose a novel Masked Joint Bilateral Filtering (MJBF) via deep image prior for digital X-ray image denoising. Specifically, MJBF consists of a deep image prior generator and an iterative filtering block. The deep image prior generator produces plentiful image priors by a multi-scale fusion network. The generated image priors serve as the guidance for the iterative filtering block, which is utilized for the actual edge-preserving denoising. The iterative filtering block contains three trainable Joint Bilateral Filters (JBFs), each with only 18 trainable parameters. Moreover, a masking strategy is introduced to reduce redundancy and improve the understanding of the proposed network. Experimental results on the ChestX-ray14 dataset and real data show that the proposed MJBF has achieved superior performance in terms of noise suppression and edge preservation. Tests on the portability of the proposed method demonstrate that this denoising modality is simple yet effective, and could have a clinical impact on medical imaging in the future.