Monitoring changes of seismic properties at depth can provide a first order insight into Earth's dynamic evolution. Coda wave interferometry is the primary tool for this purpose. This technique exploits small changes of waveforms in the seismic coda and relates them to temporal variations of attenuation or velocity at depth. While most existing studies assume statistically homogeneous scattering strength in the lithosphere, geological observations suggest that this hypothesis may not be fulfilled in active tectonic or volcanic areas. In a numerical study we explore the impact of a non-uniform distribution of scattering strength on the spatio-temporal sensitivity of coda waves. Based on Monte Carlo simulation of the radiative transfer process, we calculate sensitivity kernels for three different observables, namely travel-time, decorrelation and intensity. Our results demonstrate that laterally varying scattering properties can have a profound impact on the sensitivities of coda waves. Furthermore, we demonstrate that the knowledge of the mean intensity, specific intensity and energy flux, governed by spatial variation of scattering strength, is key to understanding the decorrelation, travel-time and intensity kernels, respectively. A number of previous works on coda wave sensitivity kernels neglect the directivity of energy fluxes by employing formulas extrapolated from the diffusion approximation. In this work, we demonstrate and visually illustrate the importance of the use of specific intensity for the travel-time and scattering kernels, in the context of volcanic and fault zone setting models. Our results let us foresee new applications of coda wave monitoring in environments of high scattering contrast.