Co-salient object detection (CoSOD) in optical remote sensing images (ORSI) is an emerging extension of salient object detection (SOD), which aims to identify common salient objects from a set of related optical remote sensing images. For this mission, we carefully construct the first large-scale dataset, CoORSI. The dataset consists of 7668 elaborately selected high-quality images and target mask annotations, covering macroscopic geographic scenes such as rivers and beaches, as well as man-made targets such as airplanes and ships, in a total of 10 categories. In addition, we propose a consensus exploration and detail perception network (CEDPNet) for cosalient object detection in remote sensing images. Specifically, a collaborative object search module (COSM) is introduced to effectively integrate high-level features and obtain inter-pixel and inter-region correlation, so as to further explore and locate collaborative objects. On the basis of this module, we design a feature sensing module (FSM), which integrates difference contrast enhancement unit (DCEU) and multi-scale detail boosting unit (MBDU) to enhance the perception of salient targets. Finally, the high-level semantic information is continuously fused with the low-level detailed features to obtain the final co-salient detection maps. Extensive experimental evaluations confirm that CEDPNet has significantly superior performance compared to other competitors in Co-salient object detection in optical remote sensing images. The CoORSI dataset, model and results will be available at: https://github.com/chen000701/CEDPNet.