If we take Turner's works seriously, it would be no exaggeration to say that society arises out of the interplay between the individual and society.
If individualism and communalism are both twin sides of society, like two shoes on feet, then subsidiarity must reach the individual and this becomes a basic tenet of Distributist family life, but also the disagreements between Libertarianism and Distributism are more nuanced. In fact I think that as the basic realities of life begin to hit home, then Libertarianism must either re-invent Distributism to stay healthy or else head in a very bad direction.
In essence I think that the flourishing of the individual and of the community are not really at odds and that each nourishes the other. Truly free individuals, seeking to preserve their freedom against both states and corporations will seek to balance and play these off eachother. Communities which nourish individuals do so with a significant element of personal liberty.
But before I begin I want to distinguish between two terms as I will use them in this post. I use freedom to refer to the Stoic concept (yes, preserved in Christianity, but not significantly different from conditions in Hinduism and Buddhism), which is an inward condition characterized by self-control and prudent wisdom. Liberty is the lack of external constraint.
Necessity of Personal Liberty to Human Flourishing
Humans are not born with significant knowledge. We learn language, develop skills, and become better at life as we lead. Learning occurs through two complementary methods, through instruction and through trial and error. One cannot learn a trade perfectly without doing it, making mistakes, and gaining the experience that those mistakes create. Thus liberty is necessary to learning how to thrive and flourish, and therefore it is necessary to human flourishing.
Liberty can be defined as "the right to make mistakes." Without genuine alternatives there is no human or social growth. Liberty is thus an absolute necessity because without it, the human spirit cannot reach more than a tiny shade of its potential and the society will be stunted or it will die. Thus it is not really the case that individualism is incompatible with human flourishing. Indeed it is a prerequisite, albeit in a tempered way.
Basics of Libertarianism and Contrast to Distributism
The basic idea of libertarianism as a philosophy is that the primary necessity in life is liberty, and that the primary condition for liberty is that the government should get out of the way aside from some specific functions, such as contract enforcement. Libertarians tend to see freedom of contract as being the fundamental freedom and it is through this that Libertarian social theory tends to be developed. Libertarians then tend to assume that things like home-owners associations are acceptable because they represent a more direct social contract between individuals and their governance. Therefore just as people can contract away their liberties in other circumstances (like business deals) so too they can do so with ultra-local government.
This is rationalized by saying that these community-level governments are not really state actors and therefore the restrictions on state actors don't apply. This distinction doesn't really make much sense to me but it is generally consistent with Libertarian thought.
Consequently one can see in Libertarian thought the idea that all social structure are essentially created by contract, and that we each have total freedom of contract in this area. At least in theory it would be theory behind the Libertarian concepts of business freedom, family structure, and the like. This is largely based on social contract theory in the tradition of Hobbes and Locke.
The Enlightenment can be seen as a philosophical offshoot of Protestantism, much the way that Distributism is often seen as a philosophical offshoot of Catholicism. Like Protestantism, it is largely hostile to tradition, individualist, and not as deeply communal as the Catholic or Orthodox traditions or the pagan traditions that came before them. In short it is an isolating approach, where individuals are robbed of traditional context and instead placed in little bubbles regarding what they believe. Secular humanism can in fact be seen as the incestuous child of Protestantism and Catholic Humanism.
Libertarians end up generally preferring the private sector to the public sector and the public sector shrinks so that its only functions are defence and contract enforcement, while the private sector increases until it reinvents the public sector. If home-owners associations are placed in the private sector merely because they are not officially arms of the government, then these terms have entirely lost any useful meaning in Libertarian theory.
To the Libertarian, then, there is nothing wrong with community ownership of roads or other infrastructure, as long as it is officially not an arm of the government, but at this point, I think this is arguing semantics and shadows instead of substance, and so the thoughtful Libertarians end up re-inventing a major part of Distributism, and resorting to ever more complex rhetorical gymnastics to prove that this is still a small government model. We will call this branch of libertarian thought "localist libertarianism." Aside from differences in labelling, we can call it a Protestant partial reinvention of Distributism. In terms of public sector projects (from a Distributist perspective) and joint community projects (from a localist libertarian perspective) these are largely identical. Where they differ are in questions about how best to help the poor.
The other main branch of Libertarian thought (what we might call the Rand camp) is simply naked liberal capitalism. Let the fire departments be privatized. Let the roads be run by private monopolies, and we can all pay or move. We will call this camp "corporate libertarianism" and they are different by virtue of the fact that they are centered around for-profit corporations owned and run for the sake of the financiers. This is fueled by the overly simplistic belief in the power of business.
Corporate Libertarianism as Unstable
Corporate libertarianism is inherently unstable. The liberal capitalist model of large financier-owned factories depends for its very existence on the disempowered worker. The real genius of servile state-type public assistance to low income workers is that this keeps these workers disempowered. Corporate Libertarianism would simply remove these subsidies and the safety net and allow the workers to negotiate for what they can. They believe wages would increase (and it is probably fair to say that nominal wages will in fact increase, but probably not enough to make up for the difference in the loss of subsidies).
The economy then becomes run by the rich, with the message that they deserve their wealth, while the poor starving family is told they shouldn't have had children that they couldn't afford and forced to depend on private charity, even if they have jobs. We've seen unbridled liberal capitalism and it isn't pretty.
The end result of course is class warfare, and the eventual rise of the very servile state the corporate libertarians claim to want to avoid. There is no other way around this.
Distributist Subsidiarity and Individual Liberty
I think it is a mistake to put too much emphasis on the communitarian aspects of subsidiarity. Subsidiarity, even in the writings of Pope Pius XI begins with the premise that society should not place a task in the hands of an organization which can be accomplished by the individual.
Subsidiarity is thus granted on the principle of general autonomy at every level, whether individual, family, organizational unit etc. The single most common violation of subsidiarity is that of micromanagement. As many employees and married people can attest, there is a large difference between being asked or even told to do something, and being first told to do something and then micro-managed as to how you do it. The first simply gives you work that you then "own" while the second gives you work but insists on still owning your work, turning you into little more than a slave or even automaton.
Subsidiarity requires trust and acceptance that different people will do things a bit differently. It is a general principle of letting individuals own their accomplishments to the extent possible and then tying those individual accomplishments together into organizational accomplishments. This is a different sense than the idea of individual liberty. Rather than grants of rights, there is a general liberty premised on the idea that one owns one's accomplishments.
Instead of a Lockean individual human rights framework, subsidiarity is a general principle of liberty, namely the right to undertake and complete works, and to own the accomplishments, honor, glory, and reputation that comes with such accomplishments. This ends up being similar to the localist libertarian liberty of contract, but ends up going in a very different direction. Distributist subsidiarity is liberty of work with the hope that it will lead to inward freedom. It is thus a deeper concept, rooted in the sacred nature of the human spirit and the work of the hands guided by the spirit.
Localist libertarianism ends up mostly reinventing large parts of Distributist social order in the "private sector" (though at this point it isn't clear what the words "private" and "public" mean). The idea of larger entities bound together from smaller entities regarding public works is present there though derived by a different path, namely by trying to push public functions into the "private sector" (whatever that means in this case), and developing contract-based ways of accomplishing this.
There are however significant problems with localist libertarianism that have largely escaped attention. Libertarians generally support a free market but aren't really very strong on government intervention in the case of monopolies. Big business is not seen as market distorting, but rather government subsidies are seen to have that role, so the lessons of the late 19th and early 20th century are lost in this group. The idea that the wealthy deserve their wealth encourages the sorts of class warfare problems that lead to the socialist servile state, or the communist servile state unless this is controlled. It isn't that there aren't mechanisms that can be used to discourage such monopolies and big business. It is that the ideology is generally against their use.
The second, and complimentary problem, is the lack of a real strategy for helping the poor. Without such a strategy everything falls apart. Currency in whatever form must circulate if a system is to be maintainable. Keeping money moving towards the top distorts and eventually causes the system to collapse. Many libertarians see property rights as a way to protect the poor but again there is a strong aversion to doing anything about property ownership. Thus there are thus key points missing from localist libertarian approaches, and while it is perhaps possible to describe a Distributist approach in localist libertarian terms, the reverse is not quite true.
 See Volokh, Eugene. "State Constitutional Free Speech Vis-a-Vis Private Landowners" and related discussions on that site.