We wish to perform a statistical study of the location and motion of individual 22 GHz water maser cloudlets, characterized by sizes that are within a few au, with respect to the radio thermal emission from young stellar objects (YSO). For this purpose, we have been carrying out the Protostellar Outflows at the EarliesT Stages (POETS) survey of a sample (38) of high-mass YSOs. The water maser positions and three-dimensional (3D) velocities were determined through Very Long Baseline Array observations with accuracies of a few milliarcsec (mas) and a few km/s, respectively. The position of the ionized core of the protostellar wind, marking the YSO, was determined through sensitive continuum Jansky Very Large Array observations with a typical error of 20 mas. The statistic of the separation of the water masers from the radio continuum shows that 84% of the masers are found within 1000 au from the YSO and 45% of them are within 200 au. Therefore, we can conclude that the 22 GHz water masers are a reliable proxy for the YSO position. The distribution of maser luminosity is strongly peaked towards low values, indicating that about half of the maser population is still undetected with the current Very Long Baseline Interferometry detection thresholds of 50-100 mJy/beam. Next-generation, sensitive radio interferometers will exploit these weak masers for an improved sampling of the velocity and magnetic fields around the YSOs. The average direction of the water maser proper motions provides a statistically-significant estimate for the orientation of the jet emitted by the YSO: 55% of the maser proper motions are directed on the sky within an angle of 30 deg from the jet axis. Finally, we show that our measurements of 3D maser velocities statistically support models in which water maser emission arises from planar shocks with propagation direction close to the plane of the sky.
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