Thermosensitive composite hydrogels that consist of a poly(acrylamide) hydrogel matrix with embedded micrometer-sized poly(N-isopropylacrylamide) microgel beads are promising models for complex, heterogeneous gels. We investigate the coupling of the microgel beads with the gel matrix and the formation of interpenetrating networks inside the microgels by confocal two-focus fluorescence correlation spectroscopy (2fFCS). This technique serves to study the effects of the heterogeneous structure of the composite hydrogels on the diffusive mobility of nanoscopic dextran tracers within the gels. Our investigations reveal that the formation of interpenetrating networks inside the embedded microgel beads depends on their cross-link density: whereas interpenetrating networks are formed inside weakly cross-linked beads, they are not formed inside strongly crosslinked beads. If the formation of interpenetrating networks occurs, the temperature-dependent swelling and deswelling of the beads is obstructed. In addition, the mobility of dextran tracers inside the embedded microgel beads is hindered compared to those in free beads and in the surrounding gel matrix. Surprisingly, the surrounding poly(acrylamide) hydrogel matrix swells inhomogeneously when the embedded poly(N-isopropylacrylamide) beads collapse upon heating. This indicates the formation of pores near the surface of the collapsed beads, offering promising means to tailor composite hydrogels for applications as membranes with tunable permeability. Our experiments also demonstrate the utility of 2fFCS to study spatially resolved diffusion in complex environments, which is of great interest in biomaterials research.