Just like any other good, money will be arbitraged for profit by moving it from where its price (i.e., its purchasing power) is low to where it is high. The arbitrage stops at the point where the purchasing power of money is the same everywhere. There is no more reason to think that gold money would not be distributed across the face of the earth according to the demand people have for it than that gasoline is distributed across the face of the earth in the same manner.
If people in a country had insufficient demand for gold money to attract gold there, then they would adopt another more common commodity like silver. The free market monetary system is one in which people are free to choose what to use as money and entrepreneurs are free to produce what money they wish. So even if there wasn’t enough gold to serve as money, it would be no argument against private enterprise money.