NYC subways were all built by the city, but most were not operated by them. There are privately built subways around the world (especially in Japan), but globally it's a rarity. This isn't a surprise, considering that subways are long term investment (and nobody can get money cheaper than a government), should be built with a long term plan to increase network usefulness (and thus the usage of each individual line) without competition between individual lines [1] and, especially if built outside already dense inner cities, should be accompanied by densification along the line to justify the investment (look at china or the danish finger plan for some examples). The operations can be private (they don't have to be, there's no clear "best way"), but the planning, financing and building should at the very least be heavily overseen by the government.
Outside of already very dense cities which heavily discourage using a car (think large asian cities, not montreal), you won't see newly built underground lines built privately. People aren't as captive today and even in the 1900s londons subways weren't very sound financially as companies.
[1]: Transit systems benefit strongly from integration between lines and modes, as well as making interchanges as painless as possible. This is mostly antithetical to multiple companies building lines inside a single city, which often ends up with little fare-integration and annoying interchanges. This can also happen with government built lines, but is less likely.