In the first case, you would bill something and get a salary out of it. If you bill x, a) your income is in dollars rather than local currency, which from experience is not great, b) you cost less to the company than US employees who cost x times k, c) you have to pay your own overhead before making it a salary. So if you bill x, your salary is less than x, and you still have to pay taxes on that salary.
If GEO or other similar arrangement, you're usually paid in local currency, presumably an amount corresponding to x at a given date, and that has extra overhead for the employer different from what a normal employee would cost.
Either way, "everyone is paid the same, regardless of location" doesn't clarify much. Thus my original question.