An accurate description of the screening behavior of asymmetric binary electrolyte solutions is obtained by applying the dressed ion theory to acquire the three leading terms in the asymptotic decay of the radial pair distribution function g(ij)(r) at large distances r. These terms give a surprisingly good representation of the distribution function for quite small r values. A minimum of two asymptotic terms are needed for representing g(ij)(r) at moderately large r in a quite wide concentration range. The relationships between the large r asymptotic behavior of charge-charge, density-density, and charge-density correlations are investigated by means of the dressed ion distribution functions. The role that the dielectric response function of the electrolyte plays for the asymptotic behavior is illuminated and the relationship between this function and the dressed ion charge distributions is given. Ionic effective charges, effective permittivities, and decay lengths and their higher order analogues are calculated for 1:2 electrolytes using the hypernetted chain approximation.