The answer is yes: take any smooth function $g_0$ on $\partial D_1$ and extend it to a smooth function $g$ on $\mathbb{R}^n$. Multiply by a smooth cutoff function $\eta$ which is $1$ near $\partial D_1$ and $0$ near $\partial D_2$. Then $f = \Delta (\eta g)$ is smooth, so in particular in $L^2$. This shows that $g_0$ is in the image of your operator $S$, and smooth functions are dense in $H^s$.

The answer is yes: take any smooth function $g_0$ on $\partial D_1$ and solve the Dirichlet problem $$ \begin{cases} \Delta g = 0 & \text{ on } D_1\\ g = g_0 & \text{ on } \partial D_1. \end{cases} $$ Now extend $g$ to a smooth function on $\mathbb{R}^n$. Multiply by a smooth cutoff function $\eta$ which is $1$ on $D_1$ and compactly supported on $D_2$. 

Then $f = \Delta (\eta g)$ is smooth and supported on $D_2 \setminus \bar{D}_1$, so in particular lies in the given $L^2$ space. This shows that $g_0$ is in the image of your operator $S$, and smooth functions are dense in $H^s$.