Serial ATA connects a storage device to a controller. Not a storage device to a storage device, or a controller to a controller.
If GbE isn't fast enough and 10GbE not an option, I can't think of anything other than copying from computer A to a fast drive (SSD on S-ATA 6 Gb/s or USB 3.0), and copy from fast drive to computer B. USB 3.0 host-to-host connections come to mind, but it is not designed for file transfer or networking, so there isn't support for those uses in OSes.