UNC has an International Friendship Program, and I suspect many of the area universities and colleges have similar programs. I have been privileged to host for a school year folks from Korea, India, China, Japan, Australia, Thailand, Hungary and other countries. The student/scholar/visiting family doesn't stay with the host, but we meet them as often as we can to bring them to our homes for a meal or go to a cultural event together. It's rewarding to host, and host families are always welcome--see oisss.unc.edu/services_programs/ friendship.html.
I am a state coordinator for Servas, a world peace and international understanding organization, a non-governmental organization recognized by the UN and established after WWII. The notion is that of building peace one friendship at a time. After an interview, you can be considered for membership. Each year you are a member, you write a letter of introduction, which opens doors to hosts around the world for short stays. The Web site usservas.org provides more details. I have had the honor of having a number of people for overnight visits, and have had instant friends hosting me in many places, including Hawaii, Canada, Scotland and Japan.