Is the second camera set up with a remote receiver like the first flash? If so then the second camera will lag behind the strobes as the master flash and slave units are able to respond in less time than the relatively slow mechanical camera operations. By the time the second camera's shutter opens the lights have already doused, and 1/250, f8, ISO 250 would need unusually bright room lighting to expose anything so a completely dark image is unsurprising.
The solution is to cascade the triggers as follows:
Transmitter set to channel 1 on hotshoe of master camera.
Receiver set to channel 1 connected to second camera's cable release socket.
Transmitter set to channel 2 on hotshoe of second camera.
Receiver set to channel 2 on hotshoe of master flash.
Slaves triggered by master flash or also equipped with receivers on channel 2.
It's possible the high shutter speed could still cause a flash sync problem, as the second camera will still expose slightly behind the first. If so then you may see the telltale sign of a bright band and dark band across the image. Normally this means the camera cannot synchronise with flash at that shutter speed, but in this case it would indicate the multiple setup has added a significant delay to the second camera's shutter opening. You could try reducing lag by locking the mirror up on the second camera, but it's less fiddly and more certain to work if you just reduce shutter speed. Ambient light is probably still too dark for a stop or two slower to register, and if not then either use less bright lamps to see your way around or stop down the aperture / lower ISO with higher flash power (if possible). As dennybeall says, you didn't specify the triggering method for the remote camera, or the model(s) of camera used. If this doesn't solve the problem, a follow up with those details might help.