Efficient transmit and receive beam management is necessary in millimeter-wave spectrum due to the inherently directional nature of antenna array transmissions in this regime. Energy efficiency of beam sweeping is further complicated when operating in shared or unlicensed spectrum, since receiving nodes are unaware if their designated transmitter has actually gained access to the medium. This article studies energy-efficient beam management from the perspective of a mobile node or user equipment (UE) that seeks to optimize the overall beam-sweeping period and the properties of each individual beam. Beam sweeping has an associated energy expenditure since the UE utilizes multiple RF chains to form the beams and incurs a processing overhead due to signal detection per beam. Two case studies are analyzed for UE beam management: downlink reception from a base station and uplink channel sensing based on directional listen-before-talk. The subsequent insights are particularly relevant for fifth-generation cellular networks that are currently being designed for operation in millimeter-wave shared spectrum.