I’m a bit confused. Why are “rigths non-material goods”? And how do you own a right? I would like to think that I have a right to own, not the other way around. It doesn’t make sense to own a right to own.
In any case, if we consider a might-is-right world we would find too many claims of ownership in relation to the total number of “objects” it would be possible to own, i.e. infinite claims of ownership and finite/scarce objects. This situation could only lead to violent conflicts since a bigger “fish” could always come along and claim rights of ownership in proportion to his might. We could say claims of ownership in relation to objects are inflated.
In a natural-right world we would find a one-to-one relation between claims of ownership and the total number of presently owned objects. This situation could easily lead to violent conflicts, but it would be possible at least to settle conflicts without violence. Why? Because any violator of a claim of ownership is trying to bring about a situation that is impossible, i.e. he is trying to inflate claims of ownership in relation to owned objects. Thus we see a potential to reach a general equilibrium in which the one-to-one relation is kept intact, and a potential to bring about the might-is-right world.
What you are asking is how this equilibrium is to come about? You are correct to point out that socialized provision of rights doesn’t really solve anything because it can only bring about, or worse keep us in the status quo of, the might-is-right world.
Now, if no person has any positive claims toward other persons how can claims of ownership be enforced against aggressors? GilesStratton mentioned the possibility to let another person, who can himself enforce a claim of onwership, homestead or buy another persons claim. Thus if big fish B robs small fish A for $100, A could sell his claim of ownership of $100 for $90 to big fish C. C would then be able to enforce his new claim of ownership against B and earn $10. The situation could also be, as you say, that no one wants to help A get the $100 back because the market price of enforcement is more than $100. Nevertheless our options are not yet exhausted. A could opt to pay more than $100, but let us say that $100 is all he has and will ever get. Well, perhaps other people, D, E and F, are in the same situation as A. Upon seeing how A has lost everything they might infer from this situation that they will be next and thus gang up with A and get his $100 from B together and for free.
The main point is that we do not need positive claims against others to enforce our rights of ownership. If poor people cannot pay the market price of enforcement they will surely gang up or be next. The probable outcome however would be that entrepeneurs would se a potential market in enforcement for poor people and would thus bring down the market price… at least A, D, E and F, if succesful against B, would have found a way to make money on enforcement.