In this article, we propose a novel self-training approach named Crowd-SDNet that enables a typical object detector trained only with point-level annotations (i.e., objects are labeled with points) to estimate both the center points and sizes of crowded objects. Specifically, during training, we utilize the available point annotations to supervise the estimation of the center points of objects directly. Based on a locally-uniform distribution assumption, we initialize pseudo object sizes from the point-level supervisory information, which are then leveraged to guide the regression of object sizes via a crowdedness-aware loss. Meanwhile, we propose a confidence and order-aware refinement scheme to continuously refine the initial pseudo object sizes such that the ability of the detector is increasingly boosted to detect and count objects in crowds simultaneously. Moreover, to address extremely crowded scenes, we propose an effective decoding method to improve the detector's representation ability. Experimental results on the WiderFace benchmark show that our approach significantly outperforms state-of-the-art point-supervised methods under both detection and counting tasks, i.e., our method improves the average precision by more than 10% and reduces the counting error by 31.2%. Besides, our method obtains the best results on the crowd counting and localization datasets (i.e., ShanghaiTech and NWPU-Crowd) and vehicle counting datasets (i.e., CARPK and PUCPR+) compared with state-of-the-art counting-by-detection methods. The code will be publicly available at https://github.com/WangyiNTU/Point-supervised-crowd-detection.