Can the PROMISE Pegasus2 R8 32TB (8 x 4TB) Thunderbolt 2 RAID System be connected into a local ethernet network to be shared off multiple computers?

Not directly – the drive is not a NAS system it has no independent ethernet or wireless interfaces. The only way you can share it is to share it through the Computing device i.e. macbook, mac pro, imac that it is connected to.