Let $\varphi$ be a function on a region $\Omega \in C^{n}$ which is locally integrable. Is it true that $\varphi = \psi_{1} \psi_{2}$ where $\psi_{i}$ are psh?
No (if I decode your abbreviation "psh" correctly, as "plurisubharmonic"). Let $n=1$. Take a function $u(x,y)=0, x<0, u(x,u)=1, x>0$. It is locally integrable, but it is not a difference of subharmonic functions. In general, subharmonic functions are sort of "almost continuous". 

