The recent rise in interest in point clouds as an imaging modality has motivated standardization groups such as JPEG and MPEG to launch activities aiming at developing compression standards for point clouds. Lossy compression usually introduces visual artifacts that negatively impact the perceived quality of media, which can only be reliably measured through subjective visual quality assessment experiments. While MPEG standards have been subjectively evaluated in previous studies on multiple occasions, no work has yet assessed the performance of the recent JPEG Pleno standard in comparison to them. In this study, a comprehensive performance evaluation of JPEG and MPEG standards for point cloud compression is conducted. The impact of different configuration parameters on the performance of the codecs is first analyzed with the help of objective quality metrics. The results from this analysis are used to define three rate allocation strategies for each codec, which are employed to compress a set of point clouds at four target rates. The set of distorted point clouds is then subjectively evaluated following two subjective quality assessment protocols. Finally, the obtained results are used to compare the performance of these compression standards and draw insights about best coding practices.