Using Finite Element computer simulations, Poisson's ratio (PR) is determined for anti-chiral structures built on rectangular lattices with disorder introduced by stochastic distributions of circular node sizes. The investigated models are parameterized by the lattice anisotropy, the rib thickness, and the radii distribution of circular nodes. Three approaches are developed. The first approach, exact in the limit of infinitely large system and infinitely dense mesh, uses only planar elements (CPS3). Two other approaches are approximate and exploit one-dimensional elements utilizing the Timoshenko beam theory. It is shown that in the case of sufficiently large anisotropy of the studied structures PR can be highly negative, reaching any negative value, including those lower than . Thin ribs and thin-walled circular nodes favor low values of PR. In the case of thick ribs and thick-walled circular nodes PR is higher. In both cases the dispersion of the values of circular nodes radii has a minor effect on the lowest values of PR. A comparison of the results obtained with three different approaches shows that the Timoshenko beam based approximations are valid only in the thin rib limit. The difference between them grows with increasing thickness.