Private gun ownership in Mexico is not forbidden, but there are limits.
Ley Federal de Armas de Fuego y Explosivos
pistolas- restricted to .380 Auto or .38 Specia; revolvers or smaller.
escopetas (shotguns)- 12 guage or smaller, with barrels longer than 25 inches, and rifles (bolt action and semi-auto.
Federal license needed for larger calibres.
The most important restriction is that there is only one sales outlet (UCAM). But private party sales are legal. The requirement that such firearms be federally registered is "widely ignored and unenforced."
Collector permits are easy to obtain. The collection is periodically inspected.

My favorite gun memory of Mexico:
Flying on the plane from Tapachula (Chiapas) to another city in Chiapas. An obviously well-to-do man in tailored jeans and Stetson, seeing that his cages of fighting cocks were carefully loaded onto the plane. He wore a beautiful set of silver-plated revolvers.