I don't think that framing of the issue is very useful on its own. Public and private entities both participate in the market, and the question of which entities are most efficient depends on which definition of efficiency one deems most relevant (i.e. which inputs and outputs one chooses to measure, and by which methods).

Also, people often attribute exclusively to government a class of dysfunctions or inefficiencies that have virtually nothing to do with an organization's ownership structure, but have a great deal to do with management structures typical of large organizations (e.g. wasteful spending due to "use it or lose it" budgets). With respect to these problems, I would be very surprised if substituting public vs. private does anything other than change minor details of how they manifest.