EDIT - because of the reshaping of the question. my answer at the bottom isn't that relevant. To the edited question: -
"Can you send data usefully over one wire, literally one wire?"
My answer is "Not with any level of success". Reasoning - if you are sending signals you need a forward and a return path for the current. If the return path is tenuously connected (such as through the ground), any ground interference in the vicinity of the return path adds to the signal received and has a tendency to corrupt that signal. However, couldn't the ground be classed as a wire? Avoiding this issue, if there were a tenuous ground connection that is validated as not adding an extra wire, the answer could be "Yes" providing the transmitted signal was powerful enough to overcome natural and unnatural interference in the vicinity of the return path.
If the signal were RF and transmitted via "a single wire" such as an antenna you could argue the answer is "Yes" but would the receiving antenna be counted as a wire making the RF argument null and void? Ultrasonic signalling or optical signalling don't need wires to make a connection so they are invalidated in my answer.
Answer to question before revision: -
I use zero-wire systems quite a lot - these receive their power and transmit data using NO (repeat NO) interconnecting wires. Data transmission is a little under 1Mb per second and of course for a (decent) network that is dreadful. Ditto for powerline comms and 1-wire and Pin&Play (worst of the lot).
If you want a decent high-speed network then wide-bandwidth systems are required and twisted pair is needed to get any reasonable distance between offices. WiFi is good but this uses sophisticated modulation techniques and radio to achieve high-speed - the ones in your home haven't got a lot of range though.
Yes you can design networking systems that only use two wires but they would be significantly faster and less-prone to errors if they were a balanced pair. You can even phantom power equipment connected to them just like 1-wire (2 really) does.
Don't be hoodwinked by all the marketing up-front blurb