Well, something does happen, but photons don't talk. It is relatively simple - each photon travels at the speed c in any reference frame. So we will ask an observer (call him Bob) who sees one photon (or rather, say a laser pulse of light, call it pulse 1) from a laser gun at a frequency $f_1$ (I know that if a pulse it won't be monochromatic, but say long enough, like 1 msec, so that it mostly is). And that observer also sees a spaceship going at c/2, in the observer's reference frame, in the same direction as the pulse, just passing him by, at exactly the same time Bob fired his laser.
Also need to assume that Bob seeing the pulse is possible, say it scatters weakly so some photons in the pulse interact with some scattering material and some photons come back to him, and he adjusts for the time of travel.
Next, 1 sec later, in that same frame, the observer sees the spaceship fire a laser pulse, pulse 2, and let's say it's arranged that it is the same type/etc laser as the other, so the pulse is pretty close to identical to the first one, and fired in the same direction. The person inside the spaceship will see it fired at the same frequency. The other observer, Bob, will of course see it Doppler shifted, at a frequency $f_2$.
Bob will see pulse 1 (let's assume it is visible, and it is visible for a distance of 1 light minute) travel for that one minute, and will see pulse 2 going at the same speed c as pulse 1, but for the full 59 seconds he sees it, it will always be 1 light second behind in distance (300,000 Kms).
So you see, photons can't see but observers who can will always see each light pulse, which is no more than a bunch of photons traveling at speed c, and one pulse trailing the other.
The observer in the spaceship, call her Alice, will see them each also (let's make the same assumptions) going at speed c, but the time will not be the same, nor are the events simultaneous for her and Bob. But it is irrelevant to this question and answer exactly what she sees. Whatever time difference she sees, it'll stay that way as one pulse chases the other, traveling at the same speed for her, and never catching up.
There is something both Bob and Alice need to watch out for. Of course neither can see the pulse of light instantaneously, so when they see it they each have to adjust the times they mark for the light travel time back to them. Since in all cases with respect to them the light traveling back to them goes at speed c, and let say they each have distance markers along the full 1 light second route, each in their frame, so they each can also see the distance markers. We could instead have recorded the times each passes each marker, and each observer use his/her markers afterwards to calculate. Different ways of observing the same physical situation.
So, to carry out this experiment you have to set things up, or figure out another way, but in all cases you see one pulse chasing the other and always remaining at the same distance. It is not a proper question what the photon would see, no observer can travel at speed c. You can of course set up a conformal reference frame, and measure things in that frame, but you will have to be in a frame not moving at the speed of light. The results are equivalent.