NIfTI and DICOM fill different niches. The NIfTI format is simple and explicit. This is ideal for science as it is easy to write software that supports this format, and they are fast to read and analyze. In contrast, DICOM is interpreted differently by different vendors, many unusual conventions are used, and personal data can be hidden into many of the crevices. I think that DICOM is ideal for archival work, but must be stored carefully. Even a thorough DICOM de-anonymization system will fail if the software that created the tags has an issue like buffer overflow (which has been seen in data from the major vendors). Further, many anonymization tools created for classic DICOM simply destroy the meta-data in enahnced DICOM. To create truly archival quality anonymized DICOM is a tremendous challenge, in particular if your IRB feels that facial features seen on T1 scans is identifiable data. 

The BIDS standard lies adds a lot of details useful for MRI processing to the NIfTI images. So a happy compromise might be to keep a private encrypted DICOM archive and share a NIfTI/BIDS dataset that has gone through a face-stripping step.