I read conflicting opinions. You've touched on an interesting controversy.
Distilled water is vaporized and recondensed, leaving it free of dissolved minerals.
Whether distilled water remains uncontaminated after processing is a small concern. If you are comparing it to tap water, you have to define the source and treatment of the tap water. Public water may have fluoride added, which is considered a positive public health issue. If you only drink bottled water, talk to your dentist about fluoridation. Public water has been treated and disinfected with chlorine, during which the bacterial content and heavy metals have been treated out.