Nobody has given a good answer yet (including me, mine was lazy), so let's get into this.

• What is "efficiency"? This is often left undefined in discussions (not always). Can we say resources are allocated efficiently or inefficiently without defining what we mean and how to measure it?

• Let's clear up the misconception that communism is always very pro-centralisation. A state will have to design its economy intelligently, which includes an appropriate use of decentralisation. To take your example of local government having budgetary discretion; it's not part of communist ideology to smack that down and say "No! Every decision is made in the capital!"

• Paper: 'Can neoclassical economics underpin the reform of centrally planned economies?*' by Peter Murrell PDF. A classic paper from 1991, highly cited. Two relevant quotes from it –

Inefficiency from planning: "With plausible values for the elasticity of substitution, he found that the efficiency loss could be as low as 1.5 percent. Desai and Martin (1983) generalized the methodology and provided time-series estimates of efficiency losses. Their estimate of the efficiency loss for 1960 was consistent with that of Thornton, but they also found such losses rising to 10 percent by 1975. When these estimates were presented, they were interpreted as a serious indictment of central planning."

Toda (1976, p. 263) examined statistical significance, and summarized his results with the same sense of paradox evinced in earlier quotations: "The Soviet institutional setting, where the industries are under various governmental regulations in acquiring the factors of production and where the price of finished goods and intermediate products are arbitrarily set, makes one suspect that the use of primary factors must be in disequilibrium. In large part, however, empirical results [examining the statistical significance of differences between factor price ratios and marginal rates of technical substitution] fail to verify our expectations."

• To bang home my second bullet point a bit more, not only can socialist economies have decentralised decisions, this can mean market-based decentralised decisions. Most successful AES countries use market-based decentralised planning (Hungary, Vietnam, China, the NEP). My very short notes on a very simple system might be illustrative, but there is a lot to say about Market Socialism: https://hexbear.net/post/282048?scrollToComments=false – This is what I meant when I said "what signals does the central government use to plan?" It's a misconception that a bureaucrat in the capital guesses what demand for fur coats will be in the provinces; they use demand as an input to the plan same as capitalism does. Even medium-sized companies in capitalism make projections of next year's demand. So just because the inputs feed into a plan, doesn't mean there are not decentralised inputs, like supply-demand figures.

• Capitalism tends towards centralisation in monopolies and oligopolies. This is intuitive because look around you. https://www.weforum.org/agenda/2016/05/joseph-stiglitz-are-markets-efficient-or-do-they-tend-towards-monopoly-the-verdict-is-in/