How would I replicate this speckled/pixelated background effect?
You might need to zoom in to see the effect fully.
This is the way which i know....
From very starting first of all
than convert this layer to smart object (its not necessary but if you need to apply any more effect easily you can apply in this)
than go to filters apply extrude this will popup a window.
In this you can reduce
size and depth if you want quick pixels block you can change this 30 to 50 or 80 to get results (i applied this filter more than 10 times to get this output you can get this within 4-5 just increase size of blocks)
blocks to pyramids make sure all option clicked as shown in the screenshot for your output, you can play with all options to discover new possibilities..
This is my output...
I have used this size to make you understand you can apply this on very small size also....
you can follow screenshot as they are to get the same result as you want..
You can go with Eric Thayne answer also, its pretty easy and clean but i doubt it matches or not with your requirements...
Hope this will help!!