From our Wikipedia friends. Countries in red have made prostitution illegal (of course, there’s as much prostitution as in other countries, but mafias are in charge wherever there is prohibition of something people really want):

What this graph shows is that other than the US, South Africa, and some Scandinavian countries, this business is outlawed wherever there are authoritarian regimes.

Color codes
Green: Prostitution legal and regulated
Blue: Prostitution legal and unregulated, but organised activities such as brothels and pimping are illegal
Red: Prostitution illegal
Gray: No data