Private tags would be used to note locations of my books around the house, to denote if a book was in storage, or at another location. It would probably serve much the same function as a "private" list, but tags are much more useful and accessible than lists. Basically private tags would be tags that I would find useful to categorize books that no one else would find useful. Users may also have tags that for whatever reason, they do not wish to share publicly.
For instance, I have a tag "SnP" which to me means that I found the book via a website called "Spirituality and Practice." I have also added other germane tags to these books, such as "Spirituality," "Work," "Social Activism," etc., that others would find useful. I don't know if the owners of the website would want their website name used as a public tag, so I would rather keep it private for just my own use.
Stuff like that...