As I understand it, UK-based pilots living in other EU countries can make some tax savings at the moment, but the whole system is currently under review and may well change.
There are many VS pilots living in other European countries. None have suggested to me that they avoid tax altogether. Most work under the arrangements for international aircrew in their country of residence, and make sure that they don't exceed the maximum stay in UK so they are not liable for UK tax. That also requires them to effectively have no ties in UK, so owning a house here can, I understand, cause problems.
Some countries, eg France, have more advantageous tax arrangements for longhaul aircrew than others, which can mean that the overall tax bill is considrably lower than in the UK. However, the process of EU tax 'harmonisation' is ongoing, and I would expect such arrangements either to become EU-wide, or to disappear altogether. You'll not be surprised which outcome my money is on...
As or 411A's story, it's very interesting, but the distances people commute within Europe are very similar to those considered normal in the US, though there is a far smaller proportion of the aircrew population doing it. Transatlantic commuters do exist, but they are very rare!