We introduce the notions of scaling transition and distributional long-range dependence for stationary random fields $Y$ on $\mathbb {Z}^2$ whose normalized partial sums on rectangles with sides growing at rates $O(n)$ and $O(n^γ)$ tend to an operator scaling random field $V_γ$ on $\mathbb {R}^2$, for any $γ>0$. The scaling transition is characterized by the fact that there exists a unique $γ_0>0$ such that the scaling limits $V_γ$ are different and do not depend on $γ$ for $γ>γ_0$ and $γ<γ_0$. The existence of scaling transition together with anisotropic and isotropic distributional long-range dependence properties is demonstrated for a class of $α$-stable $(1<α\le2)$ aggregated nearest-neighbor autoregressive random fields on $\mathbb{Z}^2$ with a scalar random coefficient $A$ having a regularly varying probability density near the "unit root" $A=1$.