X-ray computed tomography (CT) is a commonly used imaging modality in clinical practice. Recent years have seen increasing public concern regarding the ionizing radiation from CT. Low-dose CT (LDCT) has been proven to be effective in reducing patients' radiation exposure, but it results in CT images with low signal-to-noise ratio (SNR), failing to meet the image quality required for diagnosis. To enhance the SNR of LDCT images, numerous denoising strategies based on deep learning have been introduced, leading to notable advancements. Despite these advancements, most methods have relied on a supervised training paradigm. The challenge in acquiring aligned pairs of low-dose and normal-dose images in a clinical setting has limited their applicability. Recently, some self-supervised deep learning methods have enabled denoising using only noisy samples. However, these techniques are based on overly simplistic assumptions about noise and focus solely on CT sinogram denoising or image denoising, compromising their effectiveness. To address this, we introduce the Dual-Domain Self-supervised framework, termed DDoS, to accomplish effective LDCT denoising and reconstruction. The framework includes denoising in the sinogram domain, filtered back-projection reconstruction, and denoising in the image domain. By identifying the statistical characteristics of sinogram noise and CT image noise, we develop sinogram-denoising and CT image-denoising networks that are fully adapted to these characteristics. Both networks utilize a unified hybrid architecture that combines graph convolution and incorporates multiple channel attention modules, facilitating the extraction of local and non-local multi-scale features. Comprehensive experiments on two large-scale LDCT datasets demonstrate the superiority of DDoS framework over existing state-of-the-art methods.