So where are these measurements of galaxies moving faster than light?
They're redshift measurements. Check out the Wikipedia redshift article. It's good stuff.
"we can actually observe galaxies that are moving away from us at >c"
It's true. You might think it cannot be, but it can.
Um, I think I missed the groundbreaking headline that said scientists have measured a galaxy moving away from us at speeds faster than light.
It's not groundbreaking at all. It's simpler than you think.
So the first answer seems to contradict this other answer https://physics.stackexchange.com/questions/107748/how-are-galaxies-receding-faster-than-light-visible-to-observers "If recession velocity at the location of a traveling photon were greater than the speed of light the entire time the photon from a distance galaxy were traveling, we would never observe the photon".
Yes, that's a bit unfortunate, because he also says this: "galaxies with red shifts greater than ~3 were and are receding from us faster than light".
It seems like common sense that at a certain distance, space is expanding faster than light, thus preventing light beyond a certain distance from reaching Earth which means we can't directly measure galaxies traveling faster than light. But who's right?
The guy who says we can see galaxies that were and are receding from us faster than light. As per Rob's comment, see Expanding Confusion: common misconceptions of cosmological horizons and the superluminal expansion of the Universe by Tamara Davis and Charles Lineweaver. Note this: "We show that we can observe galaxies that have, and always have had, recession velocities greater than the speed of light." See page 8: "Amongst those who acknowledge that recession velocities can exceed the speed of light, the claim is sometimes made that objects with recession velocities faster than the speed of light are not observable [App. B: 9–13]."
Also have a read of the Wikipedia ant on a rubber rope article. The rubber rope is being stretched as the ant walks along it: "At first consideration it seems that the ant will never reach the end of the rope, but in fact it does (although in the form stated above the time taken is colossal). Whatever the length of the rope and the relative speeds of the ant and the stretching, providing the ant's speed and the stretching remain steady the ant will always be able to reach the end given sufficient time. Once the ant has begun moving, the rubber rope is stretching both in front of and behind the ant, conserving the proportion of the rope already walked by the ant and enabling the ant to make continual progress".
There's a section on the metric expansion of space. It says this: "By thinking of photons of light as ants crawling along the rubber rope of space between the galaxy and us, we can see that just as the ant can eventually reach the end of the rope, so light from distant galaxies, even some that appear to be receding at a speed greater than the speed of light, can eventually reach Earth, given sufficient time. However, the metric expansion of space is accelerating. An ant on a rubber rope whose expansion increases with time is not guaranteed to reach the endpoint. The light from sufficiently distant galaxies may still therefore never reach Earth".