I have zero data on this, but I am going from some anecdotal evidence.
It is entirely possible that the rank and file in private-sector unions do not support the current teacher and state worker union protests. Those I work with are pretty smart. Many are not well educated, but they have a good bit of common sense. They can see how they are affected by public-sector unions.
I see many comments on blogs and news articles that anecdotally indicate many private-sector union members recognize the problem. I'd love to see data.