The measurement of azimuthal correlations between two particles is a powerful tool to investigate the properties of strongly-interacting nuclear matter created in ultra-relativistic heavy-ion collisions. In particular, studying the near-side and away-side hadron yields associated with trigger particles can provide important information to understand both the jet-medium interaction and hadron production mechanism. We study two-particle correlations with $\mathrm{V^{0}}$ ( $K^{0}_{S}$, $\Lambda$/${\overline\Lambda})$ and charge hadrons as trigger particles of transverse momentum $8 < p_\mathrm{T,trig}<16$ $\mathrm{GeV}/c$, and associated charged particles of $1 < p_\mathrm{T,assoc}<10$ $\mathrm{GeV}/c$ at mid-rapidity in pp and Pb$-$Pb collisions at a center-of-mass energy of $5.02$ TeV per nucleon pair. After subtracting the contributions of the flow background $v_{2}$ and $v_{3}$, the per-trigger yields are extracted for two-particle azimuthal differences $|\Delta\varphi|<0.9$ on the near-side and $|\Delta\varphi-\pi|<1.2$ on the away-side. The ratio of the per-trigger yields in Pb$-$Pb collisions with respect to pp collisions, $I_\mathrm{AA}$, is measured in the near-side and away-side in the most central 0$-$10% collisions. On the away-side, the per-trigger yields in Pb-Pb are strongly suppressed to the level of $I_\mathrm{AA}$ $\approx 0.6$ for $p_\mathrm{T,assoc}>3$ $\mathrm {GeV}/c$ as expected from strong in-medium energy loss, while with decreasing momenta an enhancement develops reaching to $I_\mathrm{AA} = 2.5$ at low $p_\mathrm{T,assoc}$. On the near side, an enhancement of per-trigger yields in Pb$-$Pb to the level of $I_\mathrm{AA} = 2$ is observed. These results are compared to the published results obtained via $\pi^{0}$-h and h-h correlations in Pb$-$Pb collisions a $\sqrt{s_{_{\rm{NN}}}}$ = 2.76 TeV and they are in good agreement.