I actually worked for a Catholic company that published Bibles. It is tricky, because the paper has to be VERY thin, and a lot of printing presses cannot work with it. Of course, we NEVER considered printing out of the country, much less Red China. We had a paperback edition that we printed in-house, and our New Testament was published in-house. I believe we went with Donnelly of Illinois for our full-sized hardbound edition. our big seller. Donnelly is most known for printing telephone books, which have many of the same issues. The company who bought my former employer out has no printing facilities, but I do NOT believe they would outsource to Red China. I believe India also does some Bible printing.
I think Viet Nam and Korea print Bibles, too. Just bought my husband a Bible printed in the USA. They are not easy to find. Only two companies I know of.