In the UK, the Publishers' Association made a big fuss over one reported instance where a book from a library was downloaded in China. From the interviews they gave, you'd imagine it was a widespread occurrence, but OverDrive confirmed to me it was just the one.
The feeling here certainly seemed to be that letting people borrow books was potentially damaging the retail trade.
With most of our libraries, you have to prove that you live in the borough concerned to be able to register. Once that's done, however, things depend on what you're borrowing.
You can use your local borough's libraries (councils have a legal obligation to provide a public library service). You can also borrow from just about any library in the country, for paper books (or at least that's my understanding), via the inter-library loan service.
For eBooks, you will typically be a member of a consortium (I'm in in the borough of Hackney, which is part of the London Libraries Consortium), so there is some group purchasing power, but I suspect the areas are rather smaller than in the US.
So, I presume the consortia on service like Overdrive are buying eBooks collectively, but they're still selecting their paper books on a borough basis