An intensively demanding and open labour market exist around the world, with this as a factor, there is a very huge number of graduates go overseas after graduation for employment opportunities.
The answer to your question is yes. There are many factors, but I will only mentioned two of them. The lost of the resources to qualify a person as profesional, and the lost of possibility that a qualify person serve the country and participate in the develop of it..
Brain drain in a synonym to human capital flight of highly skilled workers. It benefits receiving countries allowing them to increase human capital stock without investment in education of those people. Clearly it has an important economic impact for both sending and receiving countries.
In early 1990s liberal reforms in the former USSR had highly negative impact on wages, especially in science, and brain drain was quite substantial. Between 1989 and 2004, about 25 000 scientists have left Russia.
High fraction of students from China do not return home; see Table of statistics here: https://en.wikipedia.org/wiki/Human_capital_flight . Also there is information about brain drain from other countries.
just to complement the answer of Yuri. I have met some chinese student of Tsinghua university who have had their master in Centrale (an elite school in France) What did they enjoy? cooking and culture. The only way to benefit from the brain drain is to make a cook from a famous university student. Moreover we have one of the most ancient written culture in the world. : hammurabi, presocratiques and stociens. With globalisation, we, Europe, have very much to offer. This can be translated in GNP and quality of life.