The Interplanetary Laser Jousting Tournament or Relativity Revised

It is said that two spaceships going away from each other at very high speeds are able to see light from the other travelling at the proper speed of light because space contracts and time slows down. What would happen if these two spacecrafts passed each other? Would space suddenly invert and time change speed? How would a spaceship travelling between two light sources, such as stars, see light from both going the right speed at the same time?

If, on the other hand, light travels at a broa...