Let $ u,v \in W^{1,1}_\mathrm{loc}(\Omega) $ and assume that $ uv \in L^{1}_\mathrm{loc}(\Omega) $ and $ u\, Dv + v \,Du \in L^{1}_\mathrm{loc}(\Omega) $. I want to prove that $ uv \in W^{1,1}_\mathrm{loc}(\Omega) $ and $ D(uv)=u\, Dv + v\, Du $.
Note: the statement is trivial when we assume $ u \in W^{1,p}_\mathrm{loc}(\Omega) $ and $ v \in W^{1,q}_\mathrm{loc}(\Omega) $ with $ p $ and $ q $ Holder conjugate (as a consequence of Hölder inequality).
Consider the interplay between Sobolev functions and absolutely continuous functions.
Assume without loss of generality that $u,v\in W^{1,1}(\Omega)$, $uv\in L^1(\Omega)$ and $uDv+vDu\in L^1(\Omega)$. Moreover, let's assume that $u,v$ are absolutely continuous for a.e. segment of line parallel to the coordinate axes and contained in $\Omega$.
Therefore, for a.e. such segment, $uv$ is absolutely continuous and by hypothesis $uv\in L^1(\Omega)$. Moreover, the partial derivatives (classical) of $uv$ exist a.e. and
$$\frac{\partial (uv)}{\partial x_i}=u\frac{\partial v}{\partial x_i}+v\frac{\partial u}{\partial x_i}\in L^1(\Omega).$$
We conclude, by using the above characterization, that $uv\in W^{1,1}(\Omega)$ and $D(uv)=uDv+vDu$.