I am curious to know what you think of unions and organized labor and what role they can or should play in an organized society. I think we all recognize that there has been a lot of bad effects of (primarily public) unions, but surely there must be some role for organized labor in a libertarian society?
Please enlighten me.
Hazlitt has a chapter that touches on this in Economics in One Lesson:
Do Unions Really Raise Wages?
The answer to the OP title is very simply: Whatever people want them to have so long as they don't infringe upon the rights of others.
As for the specific question it depends mainly upon how people organize themselves. There could be an almost entirely unionized labor force, although this is unlikely because union's tend toward inefficiency and unemployment, and at the same time there could be practically no unions. It all depends upon how individuals choose to organize themselves.
In an ideal world unions would exist only in those industries were particularly powerful in their ability to cut wages or keep wages below the marginal productivity of labor in an industry. For instance up in "mountainville", where people have a hard time moving and where there are few businesses, a union might do very well, because if the "Mountain Cereal Company" keeps wages below their marginal value then it's hard for the people to just get up and move because it's so far away from anywhere else and a new business won't start up over night.
Very much true. However it is also very plausible that a business owner may find it advantageous to bargain collectively over a mass of similarly skilled jobs rather than negotiating with each and every individual. That's perfectly acceptable and within the right of every employer to decide to require union membership as a condition of employment. It might not be the most economically efficient, but it's not fundamentally wrong.
It's for that reason that "right to work" laws, that forbid an employer from requiring union participation, are actually a statist intervention. A lot of libertarians have fallen into that trap, including Ron Paul.