Coherent perfect absorption (CPA) has been a topic of considerable contemporary research interest. Most of the theoretical treatment of CPA with beams relies on scalar (in some cases coupled mode) theories with inadequate input about the polarization states of the incoming light. In view of the lack of a full vectorial theory even for the original CPA configuration by Wan et al. [Science 331, 889 (2011)10.1126/science.1200735], we revisit the same when the incident plane waves are replaced by well-defined vector beams with or without OAM. We study the absorption characteristics of two counter-propagating monochromatic structured beams, e.g., Gaussian and Laguerre–Gaussian (LG) beams with and without orbital angular momentum, respectively, incident normally on a composite slab from both sides by fulfilling the CPA condition exclusively for the central plane wave component. We show that though perfect absorption is not achievable, there can be a substantial reduction of the scattered light. We also consider the limitations of CPA for oblique incidence and discuss the difficulties. We believe that our study will motivate and necessitate the study of recent advancements with input vector beams, retaining the full polarization information of the off-axis spatial harmonics.