This work presents an approach to multiple access for free-space laser communication (lasercom) called space-time division multiple access, which aggregates traffic from multiple users at the network edge. The objective is to share resources to lower the cost, size, weight, and power consumption per user, thereby making lasercom feasible for users that require only moderate average information rates. This concept relies on fast, agile electronic beam steering, which was implemented in this investigation using liquid crystal optical phased arrays. We designed and built an experimental terminal incorporating a bidirectional communication aperture that was shared among the users, and two independently operated acquisition and tracking apertures. Using two remote user terminals, experiments were conducted to measure access node performance for a variety of operating conditions traceable to anticipated applications. The transmit and receive directions of the downlink and uplink communications channels were rapidly hopped between the two users, and data were exchanged between the access node and a user while the optical channel dwelled on the latter. Results showed that the measured information throughput efficiency correlated well with model predictions and was high enough to realize the expected advantages in applications with many users. Throughput efficiencies, defined as the actual data throughput as a percentage of the throughput without multiple access, exceeded 85% for dwell times of 100 ms and greater. This translates into an average information rate of 400 Mb/s for as many as 20 simultaneous users. Current optical phased arrays are capable of providing fast transitions between remote users, with values measured in the range 10-18 ms. The use of persistent tracking links was a key factor in achieving fast transitions, and it was found that motion of the remote terminals had no significant impact on performance.