What's a "real-world" example of the exercise of collective bargaining rights of public employees (for example, public school teachers asserting collective bargaining rights, such as the right to unionize)? Should public employees have the right to exercise collective bargaining rights (such as the right to unionize)?

Collective bargaining refers to a process in which employers and employees negotiate with one another in an attempt to reach an agreement about the regulation of working salaries, working conditions, benefits, and other aspects of workers' compensation and rights. Employees' interests are generally presented by representatives of a trade union to which the employees belong. These negotiations normally result in a collective agreement which sets out wage scales, training, working hours, health and safety procedures, overtime, and the right to take part in workplace and company affairs. 
The outcome of negotiations between the union and employers is often referred to as a collective bargaining agreement. Both employers and employees who are members of the union who successfully brokered the agreement are subject to the conditions of such an accord.
Events in Wisconsin during 2011 provide excellent examples of the exercise of such collective bargaining agreements and their outcomes. In many instances, the agreements had outrageous consequences that not only negatively impacted those affected but also became expensive exercises to ensure that the terms agreed upon were met. A number of details concerning these events can be found at the link listed below. An additional link provides details of current California State University Collective Bargaining Agreements which should be useful.
In terms of whether you believe that employees should have the right to collective bargaining, you could use the examples provided in the above-mentioned references to determine your point of view. You obviously also need to consider the following:

Is choosing to be a member of a union a basic right?

Section 7 of The National Labor Relations Act allows employees to become members of a union and to petition for new members. Members are also protected against any action by an employer which may negatively impact on any union member, in terms of Section 8 of the same Act.

What are the advantages and disadvantages of union membership?

Be sure to check out more sources.
https://www.ueunion.org/org_rights.html

https://www2.calstate.edu/csu-system/faculty-staff/labor-and-employee-relations/Pages/default.aspx/bargaining-agreements/