I have clients in Europe and Australia that each have a Corporation (one is a Delaware Corporation, the other a Florida Corporation) that is the US arm of their company that disburses the money to their Australian and European counterparts.
Its not a cheap setup -- I wouldn't be surprised to hear both spending a few thousands a year to maintain the paperwork and whatever required.
I also have two clients that have processing offshore which required substantial documentation and one had to hire an employee in Amsterdam(?) so that they could have a European presence. Again, not cheap, but, opened up doors for them as a US based company able to take particular European payment methods.
These days, its pretty easy to set up a foreign corporation and satisfy the requirements for the merchant and processing companies. Just a matter of how much you really want to do it.
Not that there aren't people doing things illegally to do the same, but, its not impossible to have a foreign corporation that passes muster with the credit card processing companies.