Slotted ALOHA (SA) algorithms with Successive Interference Cancellation (SIC) decoding have received significant attention lately due to their ability to dramatically increase the throughput of traditional SA. Motivated by increased density of cellular radio access networks due to the introduction of small cells, and dramatic increase of user density in Machine-to-Machine (M2M) communications, SA algorithms with SIC operating cooperatively in multi base station (BS) scenario are recently considered. In this paper, we generalize our previous work on Slotted ALOHA with multiple-BS (SA-MBS) by considering users that use directional antennas. In particular, we focus on a simple randomized beamforming strategy where, for every packet transmission, a user orients its main beam in a randomly selected direction. We are interested in the total achievable system throughput for two decoding scenarios: i) non-cooperative scenario in which traditional SA operates at each BS independently, and ii) cooperative SA-MBS in which centralized SIC-based decoding is applied over all received user signals. For both scenarios, we provide upper system throughput limits and compare them against the simulation results. Finally, we discuss the system performance as a function of simple directional antenna model parameters applied in this paper.