Legally, you can employee foreign nationals based abroad as direct employees without US work authorization or SSNs but my clients always tell me that it's a complex and messy process both from a logistical and tax standpoint.
Just to put a number on that, my employer only allows foreign remote workers to be hired indirectly through a third-party staffing company, to avoid legal or tax issues associated with having direct employees in another country. The staffing company charges a fee of 40% (!) of the fully loaded compensation for this service. The fact that we're willing to pay that suggests to me that it isn't trivial to do in-house (or we're being very risk-averse, which is also possible).
On the plus side, this does mean that hiring a remote worker based in a foreign country can be solved with "just money", which is not true of U.S. immigration. If you want the problems to go away, you pay a staffing company 1.4x the compensation you want to pay the person, and the staffing company makes it happen.
