How can I achieve the faded transparency look that appears throughout this poster?
As always, there are many ways to do this effect.
Second one, easier "I Don't have time for that" way. Use photoshop, make gradient, use "cloud" filter or sponge eraser. After that just import it into Illustrator and trace it. Then you can skip it, rotate, skeeve and change colour. OR you can use it as mask (be careful as this is a complex object and Il may not mask properly or at all).