I want primarily to address the line between public humiliation and physical violence versus bullying as it exists today. How to deal with the situation from the various perspectives you ask about would vary based on circumstance and the various relationships involved, among other factors. I also apologize if I veer at all in to social services-speak, sometimes that happens on subjects like this with me.
Part of what makes Broner’s answer appealing is that it addresses head on that there won’t be some perfect world where no violence and no coercion ever take place. To some degree those things have always been and (so far as I realistically believe at least) always will be a part of human relations. Most anarchists I know seek to minimize the extent to which that is the case, and view the destruction of our society (with its power imbalances, rigid hierarchical relationships, institutionalized violence, and domination) as a good starting point for that. Therefore there is a need to attack, and to some degree there always will be a need to attack.
Specifically looking at bullying behavior, I think it is helpful to start out by looking at what bullying is. I would personally define bullying as the focused and on-going domination of an individual or a group by another individual or group through the use or threats of violence, harassment, or teasing. I would differentiate this from things like partner or child abuse or individual instances of conflict between a groups or individuals. The intention (among possible others) of bullying is to create a sense of powerlessness and isolation. Bullying could happen on the playground, it could also happen in the workplace, neighborhood, probably even the local occupation.
Given this definition, the sort of violence and shaming sometimes advocated by anarchists is distinct from bullying in that it is not about exerting power-over. In this case, bullying is more closely related to the role of police, the boss, and so forth. What gets really tricky is that in some cases those actions do look a lot like bullying – when a group is bullying an individual (hell, even if an individual is bullying an individual) it is often related to the person being bullied not fitting within a set of ideals for behavior, appearance, identity. In this case, anarchistic violence relating to dealing with behavior that violates socially agreed upon norms within a group could look a whole lot like bullying. The difference would be about degree and duration as well as whether this was an attempt to use domination to create a sense of powerlessness and isolation. I don’t think it would always be easy to draw hard and fast lines about that. In addition, if the idea of free association is taken in to account, the person being shamed, etc. is welcome to freely associate themselves with others who share their values.
My last thought about bullying as it relates to anarchy/ists: Like abusers, bullies often operate from a place of recreating either behavior that was modeled to them as the acceptable and normative, and/or domination meted to them at some point. In plain-speak, they have seen dominating behavior acted out, quite possibly on them, and are therefore acting in the ways they know. It seems quite possible (this is speculation though) that in social groups where domination and power imbalances were not the norm and were looked at as aberrations and disapprovingly, far less people would end up acting in ways that would fall in the definition of bullying I am using, though I don't believe it would ever eliminate it entirely.
I feel like this answer opens the door to a lot of questions, as well as criticisms of it. I hardly think it is perfect, and am interested to explore those further.