I understand that training this thing requires massive amount of computation, so has to be done on a massive cluster to do in reasonable time. Once they have it trained, though, what are the computational requirements like? Would it be feasible to run it on an ordinary PC?

Speaking of Go programs and computation requirements, one of the best performance hacks ever was done by Dave Fotland, the author of the program "Many Faces of Go", which was one of the top computer go programs from the '80s through at least around 2010.

He donated code from MFoG to SPEC, which incorporated it into the SPECInt benchmark. So, the better a given processor was at running Fotland's go code, the better it scored on SPECInt. Since SPEC was one of the most widely reported benchmarks, Intel and AMD and the other processor makers put a lot of effort into making their processors run these benchmarks as fast as they code.

Net result, Fotland had the processor makers working hard to make their processors run his code faster!