We investigate the microscopic structure and density fluctuations of complex coacervates of flexible polyelectrolytes using scattering of neutrons, X-rays, and light. Poly(acrylic acid) and poly(N,N-dimethylaminoethyl methacrylate) offer a well-defined model system that allows for selective labeling and systematic variation of the strength of the attractive electrostatic interactions. Two neutron scattering experiments have been carried out: (i) we use deuterated polymeric tracers in a complex coacervate with an overall neutron scattering length density that is matched to that of the solvent, to probe the conformation of single polymer chains in the complex coacervates, and (ii) we measure complex coacervates in which all polymer chains of one type are deuterated, to probe their overall structure. The single chain static structure factors reveal that both polycations and polyanions have an ideal Gaussian chain conformation in the complex coacervates. At the same time, the overall structure is similar to that of a semidilute polymer solution, with polycations and polyanions strongly overlapping to form a network with a mesh size that is much smaller than the radius of gyration of the polymers. The mesh size decreases with decreasing salt concentration, following a scaling that is in good agreement with predictions from the corresponding salt polymer phase diagram. These findings are confirmed by complementary X-ray scattering experiments. Finally, in all scattering experiments with light, X-rays, and neutrons, and for all polymer chain lengths and salt concentrations, we find a remarkable low-q excess scattering, following a power law with a slope close to -2. This points to the presence of equilibrium, large-scale density fluctuations in the complex coacervates. Dynamic light scattering experiments reveal two complementary diffusive modes in the complex coacervates, corresponding to fluctuations of the polymer mesh and diffusion of domains of varying density, respectively.