Public release of wrist-worn motion sensor data is growing. They enable and accelerate research in developing new algorithms to passively track daily activities, resulting in improved health and wellness utilities of smartwatches and activity trackers. But, when combined with sensitive attribute inference attack and linkage attack via re-identification of the same user in multiple datasets, undisclosed sensitive attributes can be revealed to unintended organizations with potentially adverse consequences for unsuspecting data contributing users. To guide both users and data collecting researchers, we characterize the re-identification risks inherent in motion sensor data collected from wrist-worn devices in users' natural environment. For this purpose, we use an open-set formulation, train a deep learning architecture with a new loss function, and apply our model to a new data set consisting of 10 weeks of daily sensor wearing by 353 users. We find that re-identification risk increases with an increase in the activity intensity. On average, such risk is 96\% for a user when sharing a full day of sensor data.