There are cases in which this does hold. Greenberg proved that if $H_1$ and $H_2$ are Fuchsian groups with a common finite-index subgroup then each $H_i$ is of finite index in $\langle H_1,H_2\rangle$. I've no doubt that this is known more generally for quasiconvex subgroups of word-hyperbolic groups, although a reference currently eludes me.

There are cases in which this does hold. Greenberg proved that if $H_1$ and $H_2$ are Fuchsian groups with a common finite-index subgroup then each $H_i$ is of finite index in $\langle H_1,H_2\rangle$. I've no doubt that this is known more generally for quasiconvex subgroups of word-hyperbolic groups, although a reference currently eludes me.

Further remark. Of course, Greenberg's theorem follows from the $l_2$-Betti number result that Andreas mentioned. But there are word-hyperbolic examples, such as fundamental groups of hyperbolic 3-manifolds, with $b^2_1=0$.