Yes, part of a living human liver can be transplanted unto another living human - the liver part will grow into a fully functioning liver withing weeks/months.
Liver transplants are done all the time. The liver is also a very versatile organ; you can transplant just one lobe of a liver, and the missing lobe will regrow on the liver from which it was taken,...
If the transplanted liver's function is exactly the same as a normal liver (i.e it's fully working), then some alcohol drunk infrequently won't do any harm (1-3 units). But a transplanted liver does...