It is not uncommon for ancient antiques to have been collected by older relatives who traveled before the value of antiquities were established. A number of these items most likely still wind up in Americans' living rooms. However, it would probably be better if they were examined and either placed in museums or returned to their home countries.
There are antique collectors all over America. I am sure that they display these pieces in their living rooms, so that all their guest may see them. Antique collecting happens all over the world just not in foreign countries. The city that I live in has tons of antique shops and I know people that collect them are proud of their finds and want them to where they can be seen by guest.
Although there will of course be some exceptions I would have thought that it is unlikely that there are many ancient antiques sat idle in the living room of normal Americans. They would be slightly more likely found in living rooms in older countries but on the whole I doubt many are in America.
People need to stop watching antique roadshow and shows of that nature. Everybody thinks they have some long lost priceless object in their house and they do not. It is usually worthless junk. Trust me, I sell on ebay for a living and am constantly inundated with people that think their stuff is worth a million dollars. it isn't.