We study the structure and stability of vortex lattices in two-component rotating Bose-Einstein condensates with intrinsic dipole-dipole interactions and contact interactions. To address experimentally accessible coupled systems, we consider Dy164-Dy162 and Er168-Dy164 mixtures, which feature different miscibilities. The corresponding dipole moments are μDy=10μB and μEr=7μB, where μB is the Bohr magneton. For comparison we also discuss a case where one of the species is nondipolar. Under a large aspect ratio of the trap, we consider mixtures in the pancake-shaped format, which are modeled by effective two-dimensional coupled Gross-Pitaevskii equations, with a fixed polarization of the magnetic dipoles. Then, the miscibility and vortex-lattice structures are studied by varying the coefficients of the contact interactions (assuming the use of the Feshbach-resonance mechanism) and the rotation frequency. We present phase diagrams for several types of lattices in the parameter plane of the rotation frequency and the ratio of inter- and intraspecies scattering lengths. The vortex structures are found to be diverse for the more miscible Dy164-Dy162 composition, with a variety of shapes, whereas for the less miscible case of Er168-Dy164, the lattice patterns mainly feature circular or square formats.