There would be no rich in a fully libertarian socialist society, as it would be extremely difficult to accumulate wealth when there is no exploitative power over others, and where anything that isn't able to be created by an individual can only be created in a voluntary worker cooperative where everyone benefits equally.
If all basic needs are freely lent out in a library economy, and everyone participating in the 2 to 3 months of yearly work equally benefitted from it in the form of free housing, food, healthcare - and public transportation and private property is abolished (distinct from personal property), then there would be virtually no avenues for an individual to accumulate enough personal property to wield any sort of substantive power over others.
difference between private and personal property
Personal property is classified as what an individual person or family can actively use themselves.
If you begin hoarding more than you or your family can realistically use, the excess is no longer considered personal property, but private property, which may get you ejected from that community if you actively hoard under a libertarian socialist society and you refuse to stop.
Quoting someone else:
I like to link the word “private” with “privation” or “deprivation”. Private property is easily identifiable by its effects on others, specifically, it’s deprivation. There are hundreds of thousands of hammers. Having one doesn’t deprive anyone of anything. At most only one person can use the hammer.
A house is usable by an entire family, and if I own it but don’t use it myself, my ownership deprives an entire family of its use. That scales to apartment buildings pretty easily. Then there’s farms where basically it’s impossible for one person to do all of the work on a farm or eat all of the products of a farm, but my ownership has the effect of depriving anyone the right to work there or the right to consume its products. A factoryn is truly impossible for one person to use, but my ownership of it allows me to deprive everyone of its products unless they meet my price demands and also allows me to deprive everyone of use of the factory to make anything at all.
Private property entails a deprivation of society of socially necessary commodities.
With the elimination of private property, and basic needs a human right, that would leave the gift economy on top of that, which would realistically be limited to just what individuals can create and share amongst themselves.