Private competition almost always ends in monopoly or trust anyway.
Private competition isn't a stable state. When two companies are competing, both are trying their best to put the other out of business. It should be no surprise at all when that eventually happens.
Monopoly, on the other hand, is a stable state--it takes extraordinary circumstances, usually government intervention, to overturn a monopoly. Even if a small competitor manages to eat a chunk of a monopoly, the monopoly company can simply buy them out in most cases. Giant corporations can even become unassailable by government, as their wealth allows them to lobby for laws that favor them.
The alternative is for corporations to form a sort of truce, where they don't touch each other's slightly different markets, effectively granting each other a monopoly in each other's areas. The clearest example of this is cable companies not laying cable in each other's territory, but there are plenty of other examples--Coke/Pepsi, IBM/Intel. This is also a stable configuration which is more sophisticated than a monopoly. The benefit to corporations is that it reduces their risk--having to overturn competitors means taking the risk of overplaying your hand--but there's no real benefit to consumers.
So ultimately, I'm not sure it makes sense to say that monopolies are the problem and not privatization. Privatization ends in monopoly.