Cecil did a column on numbered Swiss bank accounts some time back.
Basically, yes, they exist, yes, they are used. Swiss accounts had legitimate uses.
The problem is not international transfer of funds - funds don’t show up blinking red if it’s drug money or for terrorist operations, it looks like any other money.
The US (and EU countries) have leaned heavily onto Swiss and other “safe havens” (in Europe, Luxemburg and Liechtenstein) to loosen the swiss account silence and cooperate with the tax offices in the other countries / the agencies in the US tracking terrorists and drug money.
The law is of course that the Swiss are an indepent country, can make their own laws as they like and tell the others “Go f*** off”.
In practical reasons, there’s economic pressure, basically the US saying “If you don’t start cooperating with us, we will officially ban all traffic with you, let’s see whose economy hurts first”. So arrangements are made to loosen a bit and give out some information.
As for off-shore accounts: the usual draw of these is that they don’t report money to the US tax office, ideal for rich tax evaders or other shady business. This is usually done by small countries with no other means of industry, but a good technology infrastructure and close to the main centers (hence tropical islands for the US, Luxemburg and Liechtenstein for Europe).
Besides the shady business, these countries often offer lower taxes, drawing a lot of legitimate business using legal loopholes to have their main branch off-shore.
Mixing legal and shady business means that the tax office or authorities can’t simply stop everything but need to seperate carefully.