Would they also be built to produce the type of products needed by the local-ish market?
So one in Europe will have a higher fraction of diesel (used in most trucks, some cars, and some trains) compared to the USA (trucks and almost all cars use petrol).
Yes, to a certain extent. But refined products get piped all over, so it isn't a very tight coupling. Delta Airlines bought their own refinery for jet fuel. It didn't work out too well, but it was still interesting.
So one in Europe will have a higher fraction of diesel (used in most trucks, some cars, and some trains) compared to the USA (trucks and almost all cars use petrol).
(Compare: https://www.statista.com/statistics/189410/us-gasoline-and-d... - https://www.racfoundation.org/data/volume-petrol-diesel-cons... -- the ratio is very roughly reversed.)