In the context of CO₂ storage, we used ab initio-based atomic-scale modelings and simulations to study oxygen and carbon diffusion in CaCO₃ calcite. In overall agreement with earlier experimental findings, oxygen diffusion is found to take place possibly by either interstitial or vacancy mechanisms depending on the thermodynamic conditions. Contrary to almost isotropic interstitial diffusion, the vacancy mechanism strongly favors oxygen jumps within (111) planes characteristic of the CaCO₃ layered structure, with significant less-than-unity correlation factors. Our simulations show that such mechanisms cannot be applied to carbon, and complex point defects may be required to explain the diffusion of this element. While stability arguments indicate that vacancy complexes formed with CO or CO₃ missing groups may be efficient candidates to convey C diffusion, no low-energy migration path could be identified for these complexes. Focusing on the mostly stable CO vacancy complex and using generic values for unknown migration energies, kinetic Monte Carlo simulations show that this complex mechanism may be responsible for C diffusion roughly 2 orders of magnitude above its oxygen counterpart.