The structure of polyelectrolytes is highly sensitive to small changes in the interactions between its monomers. In particular, interactions mediated by counterions play a significant role, and are affected by both specific molecular effects and generic concentration effects. The ability of coarse-grained models to reproduce the structural properties of an atomic model is thus a challenging task. Our present study compares the ability of different kinds of coarse-grained models: (i)~to reproduce the structure of an atomistic model of a polyelectrolyte (the sodium polyacrylate), (ii)~to reproduce the variations of this structure with the number of monomers and with the concentration of the different species. We show that the adequate scalings of the gyration radius of the polymer~$R_{\rm g}$ with the number of monomers~$N$ and with the box size~$L_{\rm box}$ are only obtained, first, if the monomer charges and the counterions are explicitely described, and second, if an attractive Lennard-Jones contribution is added to the interaction between distant monomers. Also, we show that implicit ion models are relevant only to the high electrostatic screening regime.