Detailed molecular dynamics (MD) simulations of model single walled carbon nanotube (CNT) membranes based on atactic poly(methyl methacrylate) (aPMMA) indicate that PMMA chains significantly penetrate nanotubes through their faces. They predict very high-density values of the polymer in the interfacial area around the CNT mouths that can exceed by 50% the density of the bulk polymer at the same thermodynamic conditions. This dramatically decreases the diffusivity of relatively small penetrants (in our study, water molecules) in the nanocomposite membrane, because of the exceedingly long times needed by these small molecules to diffuse through such a dense interfacial layer before accessing the interior of the nanotubes where they can travel really fast. According to our simulations, the escape time of a confined water molecule from the blocked mouths of a CNT can exceed by several orders of magnitude the time needed by the same molecule to move through the CNT pore. Our work indicates the importance of completely avoiding (or at least minimizing) penetration of polymer chains into the CNT pores through the mouths of the tubes in enabling the efficient transport of small- to moderate-size molecules in model CNT-based polymer membranes, since this provides the highest resistance to their mobility through the membrane.