Allen asked:
Do the "free market sustaining agencies" compete with one another?
If so, how do they resolve disputes?
To answer fundamental questions yourself about any hypothetical market agency, think of non-monopoly agencies currently in existence, e.g., grocery stores. (Using this approach, you'll be able to answer most of your own questions. In addition, there is much available on the internet discussing market (anarcho-libertarian) agencies. See David Friedman, Roderick Long, and others.)
Do groceries compete with each other? Yes, usually.
How do they resolve disputes? Discussion, quid pro quo, payoffs, bring in a third party (negotiator or mediator), experiment with various solutions, take no action, etc., depending on the particular issues and the proclivities of those involved.
Best, Michael