Millimeter wave (mmW) wireless networks are capable to support multi-gigabit data rates, by using directional communications with narrow beams. However, existing mmW communications standards are hindered by two problems: deafness and single link scheduling. The deafness problem, that is, a misalignment between transmitter and receiver beams, demands a time consuming beam-searching operation, which leads to an alignment-throughput tradeoff. Moreover, the existing mmW standards schedule a single link in each time slot and hence do not fully exploit the potential of mmW communications, where directional communications allow multiple concurrent transmissions. These two problems are addressed in this paper, where a joint beamwidth selection and power allocation problem is formulated by an optimization problem for short range mmW networks with the objective of maximizing effective network throughput. This optimization problem allows establishing the fundamental alignment-throughput tradeoff, however it is computationally complex and requires exact knowledge of network topology, which may not be available in practice. Therefore, two standard-compliant approximation solution algorithms are developed, which rely on underestimation and overestimation of interference. The first one exploits directionality to maximize the reuse of available spectrum and thereby increases the network throughput, while imposing almost no computational complexity. The second one is a more conservative approach that protects all active links from harmful interference, yet enhances the network throughput by 100% compared to the existing standards. Extensive performance analysis provides useful insights on the directionality level and the number of concurrent transmissions that should be pursued. Interestingly, extremely narrow beams are in general not optimal.