Paula Stephan, Chiara Franzoni, Giuseppe Scellato have an NBER working paper entitled "Choice of Country by the Foreign Born for PhD and Postdoctoral Study: A Sixteen-Country Perspective." Here's the abstract:
We analyze the decisions of foreign-born PhD and postdoctoral trainees to come to the United States vs. go to another country for training. Data are drawn from the GlobSci survey of scientists in sixteen countries working in four fields. We find that individuals come to the U.S. to train because of the prestige of its programs and/or career prospects. They are discouraged from training in the United States because of the perceived lifestyle. The availability of exchange programs elsewhere discourages coming for PhD study; the relative unattractiveness of fringe benefits discourages coming for postdoctoral study. Countries that have been nibbling at the U.S.-PhD and postdoc share are Australia, Germany, and Switzerland; France and Great Britain have gained appeal in attracting postdocs, but not in attracting PhD students. Canada has made gains in neither.
I just want to park the link here for future reference because the paper has citations to quite a few data sources on foreign students and post-docs working in the US, the EU, Japan, Australia and Canada.
I did skim through the first part, and the overwhelming impression I get is that the number (and therefore, the share) of foreign students and post-docs in developed countries continues to grow at a fair clip, and China and, to a smaller extent, India, are the key players in this growth story.