Let $X=\mathbb{R}\times\mathbb{R}$ , we consider distributions in $\mathscr{D}'(X)$ to generalize functions $f:\mathbb{R}\times\mathbb{R} \rightarrow \mathbb{R}$
Considering partial derivatives $\frac{\partial}{\partial x} f(x,y)$ and $\frac{\partial}{\partial y} f(x,y)$ now :
For $\ T\in \mathscr{D}'(X)\ $ is it true that $\quad \frac{\partial}{\partial x} T(\phi)= -T(\frac{\partial}{\partial x}\phi)\quad $ like in the case of 'single variable' distributions ?
And if this is the case, what are the conditions that we should verify before applying such differentiation ?
As discussed in the comments, yes, this is the right way to define this, you will generally want to use integration by parts in order to decide how such an operation should be defined.
The thing you should probably verify is that $\phi$ is of the appropriate class of functions - in particular, the integration by parts step assumes that the "other" term is zero: $$ \int T' \phi dx = T\phi \vert^\infty_{-\infty} - \int T \phi' dx = - \int T \phi' dx $$
i.e. that $T\phi$ vanishes at $\pm\infty$ in some sense. For the most basic class of distributions, this is ensured by taking $\phi$ to be compactly supported (in addition to smoothness), but one can consider other classes of distributions that act on less restrictive function spaces (tempered distributions on Schwartz functions and compactly-supported distributions on smooth functions).