It depends heavily on the role and the country, but a useful rule of thumb is this: the more your job involves local customers, patients, or regulators, the more the local language matters. Healthcare, law, sales to domestic clients, teaching local curricula, and public sector roles usually demand real fluency, and often certified proficiency.
At the other end, software engineering, research, roles inside multinational companies, English teaching, and parts of hospitality in tourist areas often run substantially in English. Many expats build entire careers this way. The catch is that even in English speaking workplaces, promotion into management can quietly require the local language, because that is the language of the wider business network.
My honest advice is to start learning regardless of your role. Even conversational ability transforms daily life, widens your job market for the next move, and signals commitment that employers notice. And check whether your target country requires language certificates for visas or permanent residency, since that often matters more than the job itself.