Person detection is a crucial task for mobile robots navigating in human-populated environments. LiDAR sensors are promising for this task, thanks to their accurate depth measurements and large field of view. Two types of LiDAR sensors exist: the 2D LiDAR sensors, which scan a single plane, and the 3D LiDAR sensors, which scan multiple planes, thus forming a volume. How do they compare for the task of person detection? To answer this, we conduct a series of experiments, using the public, large-scale JackRabbot dataset and the state-of-the-art 2D and 3D LiDAR-based person detectors (DR-SPAAM and CenterPoint respectively). Our experiments include multiple aspects, ranging from the basic performance and speed comparison, to more detailed analysis on localization accuracy and robustness against distance and scene clutter. The insights from these experiments highlight the strengths and weaknesses of 2D and 3D LiDAR sensors as sources for person detection, and are especially valuable for designing mobile robots that will operate in close proximity to surrounding humans (e.g. service or social robot).