If you don't get more energy out than you put in then this is nothing more than an expensive incinerator.
The key is extracting stored energy out of the waste products ("feedstocks") in a usable form, rather than just dumping the stuff in a river or landfill to decompose and release its energy over long periods of time.
Numbers from the Missouri plant: the total inputs into the system per hour are feedstocks (122.9 million BTU) and outside electricity (3.6 million BTU). Outputs include natural gas (1.4 million BTU), light crude oil (99.5 million BTU), and carbon (6.4 million BTU). There's also additional natural gas produced that feeds back into the system.
The total energy input into the system is 126.5 million BTU/hr, while the total energy output is 107.3 million BTU, which is 84.8% efficient. (They have additional plans to use water vapor to help heating that could increase efficiency to 90%.) However, when subtracting the energy input from the feedstock, I calculate 2980% efficiency (nearly 30x) as compared to the amount of outside energy added to the system.
If you treat the carbon as a mineral output, instead of an energy output, the total efficiency is 82%, and the efficiency with regard to outside energy is about 2800%.