We investigate the structural properties of rigid linear polyelectrolytes in dilute and semidilute solutions using an integral equation theory. The Polymer Reference Interaction Site Model together with the Reference Laria Wu Chandler Closure is solved numerically taking the counterions into account explicitly. The counterions and the polymer chains, modeled as linearly connected, charged hard spheres, interact through an unscreened Coulomb potential. The pair correlation functions between the monomers of different chains, the counterions, and the monomers and counterions, respectively, are calculated for various densities and Bjerrum lengths. Based upon these quantities, the effective potential among the monomers and the counterions, respectively, is extracted. In particular, a critical Bjerrum length is determined, which separates the regime of a repulsive interaction between the counterions from the regime of an attractive interaction transmitted by the polymer chains.