Labor unions like to portray collective bargaining as a basic civil liberty, akin to the freedoms of speech, press, assembly and religion. For a teachers union, collective bargaining means that suppliers of teacher services to all public school systems in a state--or even across states--can collude with regard to acceptable wages, benefits and working conditions. An analogy for business would be for all providers of airline transportation to assemble to fix ticket prices, capacity and so on. From this perspective, collective bargaining on a broad scale is more similar to an antitrust violation than to a civil liberty.
An analogy for business would be for providers of airline transportation to vote to force all airlines, even those that don't want to join, to comply with ticket prices that the business association sets.
In other words, the key ingredient that Bob Barro misses is the element of coercion. If teachers simply got together and negotiated with school districts, but those who didn't want to join the union were free, not only not to join, which Barro advocates, but also to negotiate their own deals, then the problem with unions as monopolies would pretty much go away. Many of these people would negotiate for lower wages than the union would want, as a way of keeping their jobs and/or their flexibility. One could imagine that new hires especially, given unions' typically seniority rules, would most value their ability to negotiate a separate, and probably lower, wage.
The "right-to-work" laws that Barro discusses prevent unions from being able to force people to join and prevent unions from collecting dues from those workers who refuse to join. That's good. (Actually, it's almost good. If a business wants to deal with a union that requires everyone to join or pay dues--whether there would ever be such a case is doubtful--the business and union ought to be able to do that. That's simply freedom of association and it's something that right-to-work laws prevent.) But right-to-work laws don't undercut the ability of the union to be the sole bargaining agent. And that's the problem. As I pointed out in an earlier post, this power to be the sole bargaining agent is a power, not a right. There's no such thing as the right to make peaceful people join something they don't want to join.