MegaScience projects are unique scientific experiments and facilities of a large scale that were initiated to allow to go beyond modern knowledge in the field of fundamental sciences and open up new opportunities in the technological development. The implementation of such complex projects requires the combined efforts of many scientists and institutions at national and multinational level. The paper presents an analysis of new trends in the training of researchers in the field of elementary particle physics, high energy physics and cosmophysics for research projects and experiments at MegaScience facilities. The authors consider the vast experience of extensive training of MEPhI students for fundamental science and analyzed new approaches to organizing the individual training of researchers at the university. The result of this work was presented and discussed within the proceedings of the Third Conference “Personnel training and legal support for the implementation of scientific projects of the Mega-science class” organized by National Research Nuclear University MEPhI and Kutafin Moscow State Law University on 17 June 2021.