Placing the fines is pretty easy; they just go through their legal system, finish up the case and get their judgement. Russia has a giant outstanding fine against Google for example since Google is not censoring things the Kremlin doesn't like, even though Google has no corporate presence in Russia and the fine is iirc now larger than the entire world economy. (So it's an unrealistic amount designed to deter Google more than anything else in practice.)
The difficulty is getting enforcement; in practice, what happens is that the fine is put down as outstanding and if any executive or employee of the company enters the country, they're arrested and held hostage until the company pays up (or are held directly responsible for whatever the company is accused of). Most countries usually have corporate presency laws to avoid this sort of scenario though.
Alternatively, the judgement can be enforced through diplomatic channels, but that's a giant clusterfuck and unlikely to succeed unless it's something that's very blatantly a crime in both countries, since it's effectively retrying the case. (And even then it can depend on if the country just doesn't feel like cooperating for that specific case, for no other reason than spite; France for example is fond of doing this.)