$\mathcal D(\Omega) = \varinjlim_K \mathcal D(K\subset\Omega)$; it is the direct limit of the space of distributions with support in a fixed compact subset $K\subset \Omega$. Each of these spaces is Frechet. Thus a linear mapping $T:\mathcal D'(K\subset\Omega) \to E$ into any locally convex space $E$ is continuous if it maps convergent sequences to convergent sequences. By the universal property of the direct limit $T$ is then continuous on $\mathcal D'(\Omega)$ also.
$\mathcal E(\Omega)$ is itself a Frechet space.
Edit: Sorry, I mixed the spaces with their duals, not really awake. Commenters are right. I changed it now, and it is not an answer to the question.
SECOND EDIT: Let me try again (after my blunder before) with an explanation, along the lines: The operator is important, locally convex topologies are auxiliary.
All spaces in the question are reflexive (for the bornological topology) complete locally convex spaces. For a convenient locally convex space $E$ with (a point separating subspace of the) dual space $E'$ we can consider locally convex topologies on $E$ which are compatible with the duality (all elements of $E'$ are continuous): There is the weakest one $\sigma(E,E')$, and among all of those with the same system of bounded sets (namely the system of weakly bounded sets) there is the strongest one (the bornological one: each bornivorous set is a 0-neighbourhood). If a linear mapping $T:E\to F$ is bounded then it is continuous for the bornological topologies.
And now it comes: A linear mapping is bounded if and only if it is bounded on each sequence which is Mackey convergent to 0. (See 5.4 of the ref. below)
A sequence $x_n$ is Mackey convergent to 0 in $E$: There exists a sequence $0< t_n \to \infty$ in $\mathbb R$ such that $t_n x_n$ is bounded in $E$.
(Added in edit) Proof: Suppose that $T(B)$ is not bounded in $F$ for some bounded $B\subset E$. So there are a semi norm $p$ on $F$ and $x_n\in B$ with $t_n^2 := p(T(x_n)) \to \infty$. Then $p(T(x_n/t_n)) = t_n \to \infty$, but $x_n/t_n$ is Mackey convergent to 0 in $E$. QED.
In this sense the question can now be answered: If $T:E\to F$ is sequentially continuous, then it is continuous for the bornological topologies.
Reference: Andreas Kriegl, Peter W. Michor: The Convenient Setting of Global Analysis. Mathematical Surveys and Monographs, Volume: 53, American Mathematical Society, Providence, 1997 (pdf)