I'm the odd one out here. I do take foreign nationals without social security numbers. We are a university town and use many of the foreign students as volunteers. We work with the faculty, accepting those students who are recommended, and placing them in many many areas.
Our VP HR made the decision to move forward as each has been checked by various federal agencies before coming here, is performing well in a university setting and is championed by faculty. My other volunteers enjoy the atmosphere they bring, they quickly become "family", and he feels we are doing a little something to improve our reputation in the world.
I currently am at a low with just Nigeria, Kazakstan, Russia and Canada represented, but as far as student participation without a social - I vote a huge YES. The worst thing that has ever happened was a group that came in full native costume and brought a hibachi and food to cook for the other volunteers on their shift. Fire in the main lobby is never a good option :)