Salient object detection (SOD) in RGB and depth images has attracted increasing research interest. Existing RGB-D SOD models usually adopt fusion strategies to learn a shared representation from RGB and depth modalities, while few methods explicitly consider how to preserve modality-specific characteristics. In this study, we propose a novel framework, the specificity-preserving network (SPNet), which improves SOD performance by exploring both the shared information and modality-specific properties. Specifically, we use two modality-specific networks and a shared learning network to generate individual and shared saliency prediction maps. To effectively fuse cross-modal features in the shared learning network, we propose a cross-enhanced integration module (CIM) and propagate the fused feature to the next layer to integrate cross-level information. Moreover, to capture rich complementary multi-modal information to boost SOD performance, we use a multi-modal feature aggregation (MFA) module to integrate the modality-specific features from each individual decoder into the shared decoder. By using skip connections between encoder and decoder layers, hierarchical features can be fully combined. Extensive experiments demonstrate that our SPNet outperforms cutting-edge approaches on six popular RGB-D SOD and three camouflaged object detection benchmarks. The project is publicly available at https://github.com/taozh2017/SPNet.