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.
https://www.nytimes.com/2020/08/10/business/energy-environme...