Given that we're all about a lack of government and all about a free society, most people object to free societies by saying things like "What about X?"
Well, just like we can't exactly tell the future, us anarchists and libertarians are not superior minded entrepreneurs who can build a better society if that darned government was in the way. That would be akin to replacing the government with a new government, which is sort of awkward. What we liberty minds do actually do is convince people that people generally want things safe and secure, and so there is a market for anything a person might be worrying about not being provided.
I usually think about this in terms of two arguments: The front-end and back-end arguments. The front-end is what Murray Rothbard would use. This would be using the human action axiom (people do things in order to feel safer than before) to justify the private and voluntary actions to create businesses to provide any necessities. We have grocery stores because people tend to buy food because having food stocked in your house makes you feel safer than having to go hunt a deer every day. The other argument, the back-end, is what I imagine Robert Murphy using to justify provision of goods. This is the idea that people inherently demand something, and so there is a ready market for businesses to form, provide, and profit. For instance, most people enjoy the idea of defense. We like our houses not broken into, and we like crimes to be stopped and or adjudicated. As such, there is a market for defensive rights organizations and private courts.
Typically, when people say "but then people would just kill each other," I use the Murphyian logic to say no, people don't want to live in a world where it is normal and just to kill other people wantonly, and as such there is a market for defense and courts. In fact, most arbitrators would be fair and just to the most people because you can't very well become a successful business if you don't try to cater to as many people as possible. As such, a criminal can't just buy private courts that cater to murderers because even criminals don't want to be murdered. That'd be an inefficient business trying to hold people up if your death were a possible outcome when holding people up.
This works fine and dandy for a lot of legal concepts. Contract law, restitution law, tort law, etc. would all still hold and would become voluntary. People could create clauses in contracts with those they work with to agree to choose arbitrators who are marketing themselves as trying to be fair so as to be in the contract clauses in the first place. The one place Murphyian logic becomes murky is at Intellectual Property law. In fact, Rothbardian logic (and even Rothbard himself) is unclear about it too.
We'll start with Rothbard. In Man, Economy, and State, he justifies permanent trademarks and permanent copyrights as defense against trespass (not theft) of intellectual property. He denounces patents as being aggressive because if someone else finds a better way to do things, patents forbid this better way. It was certainly an interesting bit for me to read, having walked in as an anti-IP person thinking anyone who was Austrian would disagree with IP. Basically, the idea is that if you go around using someone else's brand name you are taking unauthorized control of the owner's production, which is trespass and not theft.
Murphyian logic would be arbitrary (no pun intended) here. This is a set of rules, and so it falls to the arbitrators (private courts) to decide if this stuff is valid or not. But would most arbitrators be for or against IP? That depends on the people they market to. Would most people be for or against IP? I disagree with IP, and I know a lot of people who call for reform (be it change of term length or change of requirements for application), but I also know plenty of people who are in favor of IP.
But why is this important? If all of this is voluntary and enforced by voluntary courts, then what does it matter? Won't people just choose for themselves and that be that? Well, no, because a free society that I would define would require definition of private property rights. There are plenty of ways Anarcho-Capitalists define property, but every anarcho-capitalist I know (Pro-IP or not) defines property as ownership over a scarce resource. Ideas (Intellectual Property) are not necessarily scarce. If I teach you how to play one of my songs, you are not taking my ability to play my song away from me. We both walk away with the idea, meaning that there is no excludability, and thus there is no scarcity. On the other hand, if I name my oranges Dash's Orange Oranges, then if you use that name at the same time I do, we both run the risk of altering the outcomes of the other's business with our own behavior, so there is rivalrousness to the good that is the name Dash's Orange Oranges.
Okay, so there's a question of IP being actual P, but still, a private justice system should make everyone happy regardless, right? Industries that worked with IP would use IP clauses in contracts, and those that didn't wouldn't. That's a problem in of itself. We can't arbitrate over arbitrary things. People have a demand for court systems that are predictable, meaning that all parties involved know generally what the processes are, and what the previous rulings are, and how the outcome is determined. If I, Dash of Dash's Orange Company take you, Darh of Darh's Oranges to court for trespass of my IP, what is the prediction? The names are obviously different, and seeing as we're competing against each other we'd disagree on the level of similarity between the names and the effects on our productions / consumers. Darh wouldn't know if it'd be a good idea to go to court with me because he cannot make an accurate prediction of how to defend himself in arbitration because there's not a quantifiable level of wrongness, and as a result he wouldn't know if he could viably resist arbitration or go to arbitration and wouldn't know the safer option. We might march toward what we have now: Giant corporations being sole providers in industry wielding long lists of IP against others.
Chaos I tell you!
As much as I like Rothbard and revere him, I disagree and think that there can be no ownership of ideas via trademarks or copyrights because of this uncertainty principle in arbitration. Many people usually retort "well nobody would create if there's no protection of ideas" but I usually say "well, Coke's formula isn't patented or anything" and "if you can't market your own product better than anyone else, you're not cut out for the role of entrepreneur." This is my inner Objectivist screaming out "there can't be arbitrary rules to human systems."
That being said, IP-like regulation would inevitably happen in free societies. You'd sign Non-Disclosure Agreements when you begin work (replacement of copyright), businesses would be doing things their own way and marketing themselves as unique (replacement of patents), and businesses lose customers when they are deceptive and would rather, through discipline of repeat dealings, have income over time rather than one off rewards (replacement of malicious trademark violation).
So maybe in the end it doesn't really matter, but it's still an exercise that divides us liberty lovers.