Often when the media describe right to work fights, they say federal regulation mandates unions must negotiate for and give benefits to non-union employees. One example from the Huffington Post is as follows, but this can be found many places.
Under U.S. labor law, a union must represent all the employees in a workplace it has unionized, even those who may not want representation. Unions say it’s only fair that all the workers in the bargaining unit pay fees to the union to cover the costs of bargaining.
On the other hand, I read in an article by the National Right to Work Committee (a lobbying group for "right to work" laws) that unions don't actually have to represent an entire workforce and are free to represent dues paying members only.
In his November 2014 majority opinion upholding Indiana’s Right to Work law, Justice Brent Dickson bluntly noted:
The union’s federal obligation to represent all employees in a bargaining unit is optional; it occurs only when the union elects to be the exclusive bargaining agent, for which it is justly compensated by the right to bargain exclusively with the employer.
I'm skeptical of the Right to Work Committee article. It's one lobbying group article against probably hundreds in the mainstream media portrayal, but the article quotes case law and a judge.
Does federal regulation actually require unions to represent non-dues paying employees? Are unions allowed to forgo their privilege as the 'exclusive bargaining agent' and negotiate only on behalf of members?