Physicist here. Many common misconceptions in the comments.

1. No, someone traveling at light speed won't arrive "instantly" or anything of the sort. It's simply not possible for massive objects to travel at the speed of light in any valid (inertial) frame of reference. Any system that does travel at the speed of light (e.g. a photon) does not have a frame of reference in which it is at rest - instead, it moves at the speed of light in all frames of reference.

If the other person travels at some speed (just) below the speed of light, the signal they send will be Doppler shifted/time dilated according to their relative velocity.

2. No, quantum entanglement cannot and never has been used to communicate faster than light. See: no-communication theorem.

Another physicist here. I see that the issue of traveling at the speed of light has already been addressed. So I’ll ignore that bit. Otherwise, yes, the time dilation would make it appear to an observer that the traveler is speaking slowly. It would also make it appear to the traveler that the observer is speaking slowly.

Where is Hank Green when you need him?

Exactly at the speed of light, the γ-coefficient would be infinite and so would be the time dialation. The eigen time of the moving person would thus be infinitely slower than the non-moving person. From the perspective of the stationary person, the time of the moving person would stand still and thus the person would never say anything. Very close to the light speed, when the coefficients are large, this problem eases but persists. The stationary person would have to wait for very long (and use a massive Doppler shift of the moving signal) to perceive something. At the end of the conversation, it will have lasted much longer for the stationary person, spending years on this. The twin paradox would basically kick in as well. If the moving person is at a speed too close to the speed of light, the stationary person might die before the conversation is over—assuming the stationary person is not immortal. That is kind of a very slow motion, yes. What a dedication, spending a lifetime on a person who can’t slow down ;-) Funny enough, from the perspective of the moving person, the effect is reversed.

I don't actually know, but to my understanding...

Firstly it would be impossible because of the doppler effect and the fact that at these speeds you would go around the earth in about the same time it takes to say "Hello" , but if you ignore that:

At a difference of around 260,000 KM/S you could actually have a time dilation difference that would be perceived by the observer (the slow one) as 0.5x and anything said by the traveler would be heard by the observer in slow motion.

Again, I have no physics background, so...

I know what your asking, but im picturing someone zooming by at lightspeed, screaming their message. The person on the ground/stationary just hears the faintest dopler effect as LS person speeds by.

Bad news: If a person was moving at the actual speed of light, from their own perspective they would arrive at their destination instantly. This means they wouldn’t have time to send or receive a message at all!

Assuming a velocity close but not quite as fast as C, yes, you would see severe differences in the speed of the communication. One party would be super slowed down and the other would be super sped up.

Anything moving at the speed of light in one reference frame is moving at the speed of light in every reference frame—including its own.

Which is to say, it’s not a real reference frame at all—the experience of moving at the speed of light would be instant teleportation with no subjective elapsed time. So trying to talk to someone moving at light speed would be like talking to a still image.

Yes. Distant galaxies that are moving away from us at relativistic speeds exhibit measurable time dilation in their inner workings.

How would you even measure time dilation in a distant galaxy? Consider standard candles like 1a supernova, which explode with near uniform power. These supernova can be observed from intergalactic distances. Gather data and record the times for various supernova explosions. You'll find that the same types of explosions take longer in more distant galacies, and that the extra time is exactly what relativity predicts.

So, I would assume that you would be communicating through radio waves. If an object broadcasting a signal moving at the speed light away from you, I would further assume you experience a severe Doppler effect. To the point that I don't think you would experience anything coherent. You would receive small packets of information at a time, scattered across several million years.

This is just my initial impression on the fly, do not take this as any sort of gospel. I also did some communcations work for a time. So, this is tickling my brain and I might spend the rest of my evening in my books.