Hmmm, you are in an international forum and don't say where you are from so I will assume you are "American".
What type of bike do you want to buy? New or Used? I assume that you want it plated for street use?
There are ways around a lot of things and anyone correct me if I am wrong as this is just from personal experience. Unless you have a DNI number for Chile, you can not buy a new bike there. You can buy a used bike but can't plate it. You can not use someone else's bike to cross borders. You can buy a bike from another gringo but it is sometimes dificult to cancel one person's permission and initiate a new one for the new owner.
It is worse in Argentina
. Better in Bolivia and Peru but the easiest and cheapest place is Paraguay.