X-inactivation is a little bit more complicated than that. While the process of X-inactivation initiation is random, once a cell has settled on one chromosome, all its daughter cells will silence the same chromosome. The initial process happens in the early embryo, so large patches of the body have the same X chromosome silenced.
This pattern is visible in some animals. E.g. a tortoise cat’s pattern arises due to the hair color gene existing on the X chromosome. Consequently, male tortoise cats are rare (XXY, XXXY etc only)