We studied the structural, electronic, and magnetic properties of Mn-doped topological insulators Bi₂Se₃ and Bi₂2Te₃ within the density-functional theory (DFT) using the generalized gradient approximation (GGA) in the framework of the fully relativistic spin-polarized Dirac linear muffin-tin orbital band-structure method. The x-ray absorption spectra (XAS) and x-ray magnetic circular dichroism at the Mn K and L_2,3 edges were investigated theoretically from first principles. The calculated results are in good agreement with experimental data. The complex fine structure of the Mn L_2,3 XAS in Mn-doped Bi₂Se₃ and Bi₂Te₃ was found to be not compatible with a pure Mn³⁺ valency state. Its interpretation demands mixed valent states. The theoretically calculated x-ray emission spectra at the Mn K and L_2,3 edges are also presented and analyzed.