We completely characterize $Δ$- and local subexponentialities of positive-half compound Poisson distributions and extend the characterization on two-sided distributions. Moreover, $Δ$-subexponentiality of infinitely divisible distributions is characterized with new conditions, and local subexponentiality is newly characterized in the two-sided case. In the process closedness properties of these subexponentialities are derived, particularly for distributions on $\R$. Most results are obtained by exploiting monotonic-type assumptions. We apply our results to distributions of supremum of a random work and a randomly stopped iid sum.