Since the discovery of the quantum anomalous Hall effect in the magnetically doped topological insulators (MTI) Cr:(Bi,Sb)$_2$Te$_3$ and V:(Bi,Sb)$_2$Te$_3$, the search for the exchange coupling mechanisms underlying the onset of ferromagnetism has been a central issue, and a variety of different scenarios have been put forward. By combining resonant photoemission, X-ray magnetic dichroism and multiplet ligand field theory, we determine the local electronic and magnetic configurations of V and Cr impurities in (Bi,Sb)$_2$Te$_3$. While strong pd hybridisation is found for both dopant types, their 3d densities of states show pronounced differences. State-of-the-art first-principles calculations show how these impurity states mediate characteristic short-range pd exchange interactions, whose strength sensitively varies with the position of the 3d states relative to the Fermi level. Measurements on films with varying host stoichiometry support this trend. Our results establish the essential role of impurity-state mediated exchange interactions in the magnetism of MTI.
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