France at least appears to be interested, requiring you to explicitly list every foreign place you can stash money on tax returns every year. If you have any kind of French rental income, no matter where you are in the world, I understand you have to declare. There is an EU directive for reciprocal information sharing and I understand it is used.
There are fines and penalties for not declaring a foreign account. The fine is not income or wealth linked, so it seems like it wouldn’t really bother anyone with the sort of wealth to benefit from not declaring; the penalty is having the stashed money treated as income for the year it is discovered, which seems like it could make certain people nervous.
> Note that it has been a nightmare for US people living in EU
It appears to make life much harder in all manner of ways. When I opened my French bank account, the forms placed a heavy emphasis on me not being a US citizen. I asked and was told that it would be much more difficult to open an account.