Deep learning-based image enhancement has significant potential in the field of ultrasound image processing, as it can accurately model complicated nonlinear artifacts and noise, such as ultrasonic speckle patterns. However, training deep learning networks to acquire reference images that are clean and free of noise presents significant challenges. This study introduces an unsupervised deep learning framework, termed speckle-to-speckle (S2S), designed for speckle and noise suppression. This framework can complete its training without the need for clean (speckle-free) reference images. The proposed network leverages statistical reasoning for the mutual training of two in vivo images, each with distinct speckle patterns and noise. It then infers speckle- and noise-free images without needing clean reference images. This approach significantly reduces the time, cost, and effort experts need to invest in annotating reference images manually. The experimental results demonstrated that the proposed approach outperformed existing techniques in terms of the signal-to-noise ratio, contrast-to-noise ratio, structural similarity index, edge preservation index, and processing time (up to 86 times faster). It also performed excellently on images obtained from ultrasound scanners other than the ones used in this work. S2S demonstrates the potential of employing an unsupervised learning-based technique in medical imaging applications, where acquiring a ground truth reference is challenging.