This work studies the net sum-rate performance of a distributed reconfigurable intelligent surfaces (RISs)-assisted multi-user multiple-input-single-output (MISO) downlink communication system under imperfect instantaneous-channel state information (I-CSI) to implement precoding at the base station (BS) and statistical-CSI (S-CSI) to design the RISs phase-shifts. Two channel estimation (CE) protocols are considered for I-CSI acquisition: (i) a full CE protocol that estimates all direct and RISs-assisted channels over multiple training sub-phases, and (ii) a low-overhead direct estimation (DE) protocol that estimates the end-to-end channel in a single sub-phase. We derive the deterministic equivalents of signal-to-interference-plus-noise ratio (SINR) and ergodic net sum-rate under Rayleigh and Rician fading and both CE protocols, for given RISs phase-shifts, which are then optimized based on S-CSI. Simulation results reveal that the low-complexity DE protocol yields better net sum-rate than the full CE protocol when used to obtain CSI for precoding. A benchmark full I-CSI based RISs design is also outlined and shown to yield higher SINR but lower net sum-rate than the S-CSI based RISs design due to the large overhead associated with full I-CSI acquisition. Therefore the proposed DE-S-CSI based design for precoding and reflect beamforming achieves high net sum-rate with low complexity, overhead and power consumption.