In this paper, we consider the question of the complete faithfulness of the $p$-free quotient module of the dual Selmer groups of elliptic curves defined over a noncommutative $p$-adic Lie extension. Our question will refine previous questions on the complete faithfulness of dual Selmer groups. We also consider the question of the triviality of the central torsion submodules of these Iwasawa modules and we see that this latter question is intimately related to the former. We will also formulate and study analogous questions for the dual Selmer groups of Hida deformations. We then give positive answer to our questions, and establish "control theorem" results between the questions in certain cases.