Private vs. Public is not the discriminating point, centralization is.
The Soviet Union went down because it represented an attempt at strictly centralized management, coupled to punitive information management where any feedback was assumed to be dissent and swiftly repressed.
As you may know, negative feedback systems are the key to stability and USSR is a case study of everything the opposite.
China is hardly a legit market economy, more of a compromise to afford some autonomy of judgement and decentralization.
