Examples of undecidable problems whose intersection is decidable
So here is a example, which is probably not as nice as you wanted it to be, but less trivial than the one you have mentioned.

Let $L_1,L_2\subset \{a,b,c\}^*$ be two undecidable languages, and $L_3\subseteq \{a,b,c\}^*$ a decidable language. We define

\begin{align}
L_A&amp;:=\{a\,w \mid w\in L_1\} \cup \{c\,w \mid w\in L_3\}, \\ 
L_B&amp;:=\{b\,w \mid w\in L_2\} \cup \{c\,w \mid w\in L_3\} .\\ 
  \end{align}

Clearly, both $L_A$ and $L_B$ are not decidable, however their nonempty intersection
$$ L_A\cap L_B =\{c\,w \mid w\in L_3\}$$
is decidable.