Preface: I've always thought communism got too much hate in non-communist circles, especially since everyone but a pure anarcho-capitalist supports nonconsensual redistribution of wealth in some situations. I embrace the legitimacy of private property, but the pragmatic comparison between communes and markets is a lot less clear than market advocates believe.
A popular pro-market argument:
In a free market, the way to get rich is to provide something others want.
It's true, it's a good point. But there's a corollary: if you benefit from solving others' problems, then you also benefit from them having the problem in the first place. So markets also incentivize causing the problem your business solves, providing only temporary solutions that make people dependent on your service in the long run, and discourages doing anything that would help them for free (like a positive externality) because that deprives you of the opportunity to profit from it. A commune eliminates all of these incentives, not just the good ones.
As another way of looking at the incentive argument, market advocates will (rightly) argue that communism allows people to live on the work of others without producing any value themselves. But there are two factors being overlooked here that would greatly mitigate the problem.
First, reputation. Reputation is a powerful force that even market advocates appeal to frequently when talking about how bad business practices are disincentived; it also applies in communes. Even if you're entitled to food and other necessities without working, social boycotting is powerful, and even without outright boycotting, our emotional need for peer approval is very powerful.
Second, work, for many people, is more or less enjoyable. I'm a programmer and have put many hours into contributing to open source projects, for which I get zero pay and often no personal benefit. I do it for emotional reasons; I derive pride from contributing even outside of other people seeing it. These forces are extremely powerful, and they would mitigate freeloading in a commune.
Of course, the work-is-enjoyable argument doesn't apply equally to all occupations. I doubt someone who works in construction finds their work as enjoyable as I find mine. And this also doesn't help coordinate work where it's most needed. But it certainly does mitigate the problems, and people who hate communism tend to overlook it and argue as if these factors only apply in market societies.
Concentration of power
Market advocates bend over backward trying to defend markets from the monopoly criticism. Many ancaps will go as far as to argue that monopolies can't happen in a free market. Although they're right that most monopolies and oligopolies are created by government red tape and favoritism (and that government itself is a monopoly), and there are lots of good arguments to make that monopoly abuse is not a critical problem with free markets, I've never heard a convincing argument that markets don't tend toward the concentration of power at all, since economic power facilitates its own growth and things like vendor lock-in exist. Monopolies can form in a free society, and they can abuse their power, especially by acting in secrecy. Communes don't have this problem.
Property enforcement difficulty
How much of an issue this is depends on your beliefs about the nature of property rights. If you share mine, particularly the half-defense of intellectual property and the value-splitting stance on natural resources, well, there's a problem: as sound as those might be axiologically, they're basically impossible to enforce because both depend on quantifying subjective experience and intellectual property is trying to fight against the nature of physical reality. Note that having unenforceable property is worse than abolishing property in this way: unenforceable property allows your income to be lost; in a communist society, your income is effectively fixed.
Of course, this is a much lesser issue if you're a typical ancap who accepts homesteading and rejects intellectual property altogether.
Finally, market advocates are always on about how enterpreneurs are brilliant and deserve to be rewarded for their wise leadership; how the market rewards good decisions and punishes bad decisions. But they downplay how much luck is involved. Even if you assume there's a real difference between luck and good amoral decisions (I actually think that's a very dubious distinction), a lot of market success is luck. The environment you're born into is a huge factor. Even after that, you never have all the information relevant to your decision, so it's wrong to characterize markets as pure meritocracies.
Accordingly, in a market society you can lose everything and be plunged into debt without making any bad decisions. You can suffer all sorts of natural or financial disasters. And things like mutual aid societies can mitigate this, but then those aren't really a market construct so much as a mix-in of voluntary communism.
So to recap: markets are cool, but they aren't the perfect solution to all problems that libertarians like to think, and communes seem quite competitive with them pragmatically.