A nitpick here, but "visa" gets tossed around a lot. A visa is a document which allows you to request permission to enter a country for some purpose. To work in a country you need a work and/or residency permit. Even in the US, there are situations where you can work with a an expired visa, provided your other immigration documents are current. But if you leave, you need a new one. Similarly UK citizens can still work in some European countries without a visa, because there is no requirement for them to enter with one (but the officer at the border might still refuse entry).
Again, depends on the country. Switzerland (which I appreciate is not in the EU) does not require visas for many "friendly" 3rd countries even if the person is coming to work.
The UK is considered non-EU so there would need to be a visa for a Brit working in the EU.