Let $G$ be an infinite finitely generated group and $S$ a generating set. Define density with respect to the sequence of balls $S^n$.

If $H_1, H_2 \leq G$ have positive density, must $H_1 \cap H_2$ be (a) nontrivial? (b) positive-density?

Obviously if $H_1$ and $H_2$ have finite index then so does $H_1\cap H_2$, so such subgroups are not relevant. It is possible for an infinite-index subgroup to have positive density, e.g., $F_2 \leq F_2 \times \mathbf{Z}$ (with the "summed" generating set). The group $G = F_2 \times F_2$ is a near miss: the two factors don't have positive density in $S^n$, but they do have density about $1/n$, which is fairly large.