The short answer is yes*, with emphasis on the asterisk. Data that may be considered sensitive can certainly be published openly, without restrictions on access and/or reuse, but it will likely require some level of modification. This page provides an overview of some best practices and conceptual guidelines for how to do so.
Reproducibility is one of the primary motivators for funders and publishers to enact data sharing requirements - without underlying data, it is often very difficult or impossible to reproduce the results of a study without having to conduct a repeat experiment. In many instances, more data are collected than are analyzed, and sharing those data can be useful for inspiring future work, but if data sensitivity is a concern, sharing only as much data as are necessary to reproduce a given study should be shared publicly.
In some instances, far more data and metadata are collected than are analyzed or presented in an associated article. For example, exact timestamps are often collected for online surveys (both inside and outside of research contexts), but this data is often not actually analyzed, especially when the time window of data collection is narrow (e.g., a week). Sometimes, a dataset's sensitivity can be greatly reduced or entirely eliminated by removing variables that are not directly analyzed in associated outputs.
In a related vein, data are sometimes collected in one form but are then transformed or otherwise modified into a more generalized form for analysis (e.g., binning precise age in years into 10-year age bins). In circumstances like this, sharing only the more generalized form of data publicly can still ensure reproducibility while also limiting risk of exposing sensitive data.
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