Let $D$ be the (open, closed or whatever) unit disk and let $T\colon \mathbb R^2\to\mathbb R^2$ be a rotation or trnslation such that $D=A\cup T[A]$ for some set $A$ with $A\cap T[A]=\emptyset$.
If $T[\partial D]=\partial D$, the point $0$ must be a fixed point of $T$; but then $0\in A$ implies $0=T(0)\in B$ and vice versa, contradiction.
Hence $T[\partial D]\ne \partial D$, i.e. these two circles intersect in two distinct points $a,b$.
Let $C_a$, $C_b$ be the circles through $a,b$ around the center of rotation (whch is on the line $ab$) - or in case of a translation the lines orthogonal to $ab$ through $a, b$, respectively.
These circles/lines intersect (not touch) $\partial D$, hence chop the interior of $D$ into three parts with nonempty interior. For the two "outer" parts we have that their images under $T$ are disjoint from $D$, hence thy must be $\subseteq B$. But they are also disjoint from $T[D]$ hence must be $\subseteq A$, contradiction.