Carracks were used all over the European shipbuilding world, including in the Mediterranean and the Baltic, and were also built in the Americas after the Spanish and Portuguese conquests there.

So the answer to your question almost has to be a very vague "yes, all of the above," unless you can narrow down a more specific area of the globe or a time period.

One thing to keep in mind is that in the late Middle Ages, the distinction between royal and private yards is very fuzzy; even royal shipyards such as Portsmouth in England may have been built with the backing of a monarch but employ many private contractors or men and women doing piecework; they are not analogous to modern commercial enterprises. Carracks were ubiquitous enough to be used in the merchant service, in royal- or nobility-backed expeditions such as Columbus' voyages of discovery, and also as purpose-built warships, so a mixture of building places and techniques were present.