Can international graduate student take the free online courses offered in Coursera? Is it necessary to notify to the graduate school when taking the free online courses?
As always, it's better to check directly with your advisor, who is in the best situation to answer exactly your question. That being said, as Zenon said, you should be able to do pretty much what you want during your free time, including taking online courses. However, if you take so many classes that it prevents you to work normally for your grad school, then it might be a problem. Furthermore, if you expect the online courses you take to count for your graduate program (i.e., to replace other courses you would normally have to take at your school), then only your advisor can answer that.