Let $F$ be a free group of rank $2$. If the intersection triviality problem is decidable for $F\times F$, then using the Mikhailova construction, it would be decidable given a finitely presented group $G=\langle X\mid R\rangle$ and a word $w$ in $X$ whether $w$ has infinite order in $G$.  This latter problem was proved undecidable in  G. Baumslag, W. Boone, and B. Neumann, Some unsolvable problems about elements and subgroups of groups. Math. Scand. 7 (1959), 191–201.

Indeed, embed $F_X\times F_X$ in $F\times F$.  The subgroup $E=\{(u,v)\in F_X\times F_X\mid u=v\ \text{in}\ G\}$ is finitely generated.  It intersects $\langle (1,w)\rangle$ trivially if and only if $w$ has infinite order in $G$. 

So intersection triviality is undecidable for $F\times F$, although this group has decidable word problem.

I assume this is what the OP was getting at in his/her comment.