The Aurivillius layered compounds with predominant fraction of paramagnetic transition metals are an emerging playground for discovery of magnetoelectric or multiferroic compounds. This aim was recently achieved by incorporating Fanions in the perovskite layer of the (Bi2O2)(CoF4) compound, only described so-far in a disordered model (unit cell : ap, ap, c). Here we report the investigation on the representative compounds (Bi2O2)(MF4) (M = Fe, Ni) using single crystal, synchrotron, neutron and electron diffraction. These reveal that the crystallographic average cell (2xap, 2xap, c) is orthorhombic, polar and accompanied by versatile (in)commensurate modulations. The supercell model was fully refined for (Bi2-xO2)(FeF4) (q = (0, ½, 0)) in the P2111 polar space group with transverse Fe-displacements relative to q. Bi-deficiency is compensated by a mixed Fe 2.5+ valence, but, the ideal stoichiometry is preserved for (Bi2O2)(NiF4). Both compounds are antiferromagnetic below TN = 89 K (Fe) and 45 K (Ni) with moments lying in the (ac) plane and a weak ferromagnetic component along the b-axis. DFT calculations validate a strongly anisotropic distribution of magnetic exchanges (Jab/Jc > 10). A broad anomaly on the dielectric constant at TN and a polarization loop at room temperature were obtained on (Bi2-xO2)(FeF4) single crystals, revealing multiferroism with magneto electric couplings.Read less <