In my story, I have two huge generation ships racing each other to another star system. Assume they are on an exactly parallel path, neck-and-neck, about one astronomical unit apart. They are traveling at exactly the same speed - point eight cee. They know the other ship is there, somewhere. They launched at the same time, although from separate in-system facilities.
The form of drive is important. They are traveling in a modified Alcubierre drive - that is, using the vernacular, they are in a space-time bubble that preserves relativistic effects within the ships. The drive creates a low-pressure space-time area in front, and a high pressure space-time area behind. The drives also interplay with the Higgs field to reduce the effects of inertial mass. Thus, the bubble is effectively 'surfing' through space, being pushed by the high pressure from behind. Sort of like an airplane stays in the air. There is no intention to use the system to go faster than the speed of light, just close to it.
Because of this, the ships are huge. They have huge fusion reactors, and a complete physics laboratory. Power is not an issue. They each carry tens of thousands of people, and have academic institutions that train new physicists born on the ships. Think in terms of a complete city. Knowledge and the ability to build new equipment is not a problem.
The question is, traveling at such extreme speeds, wrapped up in a bubble, can they communicate with each other? If so, then how?
Because they are adversaries, there are no entanglement possibilities between them.
Some additional background to consider - light takes about 500 seconds to go one AU. The trick is not in the relative speed between the ships, but that they are both traveling at point 8 cee relative to where the light was first emitted. That is, there would be a Doppler effect as the light arrived at the other ship.
The point of protecting the inhabitants from relativistic effects is that I sincerely doubt biological processes could function in the relativistic effects of point 8 cee.
For consideration: Would there be a null point in the bubble between the front negative pressure and the rear positive pressure that could be used?
A point of physics. When you project a beam of light from a spaceship, the point of origin effectively drops dead. That is, the beam origin does not gain or loose momentum from the spaceship. At the point in space-time where you 'drop' it, it stays there. Perfectly dead in the - well - space-time. The ship continues on. It does not 'follow' or 'trail' the ship. If you beam it behind you, then you travel away from the originating point of the beam, the point of origin just hangs there, and the speed of the light is determined relative to that 'dead drop'. Not the speed of the ship. Not the speed of anything around it. If it is a completely uni-directional laser beam, and it comes out the extreme rear of the ship, the ship occupants will never be able to detect it unless they come around. Baring gravity, it goes in a perfectly straight line from the original point of origin in space-time.
Please Note I use point 8 cee because c is constant in all inertial frames, but the speed of a spaceship certainly isn't. I use cee as a unit of speed, as in miles per hour.