It is theoretically possible to have multiple computers connected to a single SCSI device, but the computers AND the device all need to be aware of the multiple computers. (This also carries over to FireWire.)

As far as I know, no 'conventional' hard drives support multiple computers at the same time. Some ultra-high-end server drives did, but they required that the SCSI cards in all connected computers support it, too. This role has been effectively replaced with FibreChannel, which can do it through the use of a FibreChannel switch.