In specific to the Jeff Galloway Run/Walk method. The point of the method is to ease the stress that running has during training and races. It splits the race up in to running segments and walking segments with the idea that the walking segments facilitate recovery which allow the running segments to be faster and easier than if you just ran the entire course. Each walking segment is deliberate and planned. The runner doesn't run up to the point of exhaustion and is forced to slow down. Instead, the runner slows down to a walk well before that.
How fast should the walking be?
A leisurely pace. Walking segments typically are not that long to begin with. The point is to give some rest period for recovery. Though, it won't necessarily hurt to speed walk if you can spare it.
How fast should the running be?
During training, you should be running at a "conversational pace". Meaning, you should be able to hold a conversation with somebody while running. A common test (if you're in the U.S.) is the "Pledge of Allegiance" test where you recite the Pledge of Allegiance. If you can't finish a sentence without gasping for air, then you're going too fast. Faster, more fit people, can do this at a faster pace, but it's always relatively easy.
During a race, you don't necessarily have to change the tactic. Especially if you're only competing against your own time. A training schedule should have a period of rest for you body to overcompensate. Plus all the adrenaline and people around you. It all combined generally would push you to have a faster "conversational" pace. (As a side note, it's never a good idea to do something dramatically different on race day.)
Does it change for half-marathon or full-marathon?
The running and walking paces do not change for these. The lengths of the segments may change depending on the training requirements.