Relativity just means that the laws of physics are the same for all observers that move with a continuous velocity (no matter if positive, negative or zero). Which means there is no absolute frame of reference and every observer could define their own inertial frame of reference in which they are the ones standing still and the rest of the world moves.
So say you have 2 spaceships in pitch black void with no known reference. Then it would be possible to determine the relative velocity, that is whether the ships move towards each other, away from each other or in unison/standing still. But you couldn't tell if that is because you are moving or because they are moving or because both of you are moving or one faster and one slower.
Both of you would have the impression that you are standing still and the other is doing the movement.
So no 2 observers can be in the same inertial frame of reference and it's an interesting thought that if a consciousness spreads beyond a point-like space how that would be effected by special relativity. However in practical terms the relative movement between these frames of reference is quite small or even 0. Like think about what it would mean if your body parts moved with relative motion that is relevant to special relativity. It would literally tear you apart and in that case consciousness would be your smallest problems. So as "you" have little to no relative speed to the rest of your body you're consciousness is continuous and any delay is "business as usual".
And with respect to the consciousness of other people it's the same problem. Like the way you'd interact with other people's consciousness is either by seeing touching or speaking with them. In any case the transmission of information happens at speeds like the speed of sound (340m/s for speaking) or the speed of light (300,000,000 m/s for seeing and probably for the neural transmission after touch). So if your two observers stand next to each other than their relative speed is small or zero compared to these numbers than it doesn't really matter all that much. And if one observer speeds away with a relative speed of 300m/s communication would actually be difficult by speech. Simply because over 300m the signal would have been damped to 0.
For light though the signals could still be transmitted however if the information transmission speed and the relative speed are on the order of magnitude then the information will reach you way later. Think of "seeing" as sending a beam of light out, the beam hitting the target and getting reflected and coming back to your eye. Now if the thing your looking at moves away with a relative speed of light, the light from your eye will take much longer to hit it and then has to go in return as well so what you see when you see it isn't where the person is now but where they were when the light had reached them.
But that doesn't mean that the other person is distorted or "living in the past", it's just a problem of when you get aware of where the other person is.