random copied and being truly random
The random number generator in AE seems to be linked to the layer index, but generates the same results for every composition. To demonstrate here's a comp with a shape layer where the vertices of the shape are generated by the wiggle() function:



Here's the expression:

var newPath = [];
for (var i=0; i&lt;=160; i++){
    x = i*thisComp.width/160;
    y = transform.position.wiggle(1, 100, octaves = 1, amp_mult = .5, t = i)[1];
    newPath[i] = [x,y];
}
createPath(points = newPath, inTangents = [], outTangents = [], isClosed = false)


if I duplicate that layer in the same comp I get two different paths (I've given each a different colour, but the expression is the same):



If I add layers to the comp before the shape layer, so that the shape layer's index changes this also changes the shape of the path. You can also explicitly set the index of the layer generating the wiggle, e.g.:

var i = math.floor(effect("Slider Control")("Slider"));
var w = thisComp.layer(i).transform.position.wiggle(1,2);


So the index of the layer is used to generate random values for the wiggle() function. Quod erat demonstrandum.

This means that if I duplicate my comp, each duplicate comp produces the same result. Here I've duplicated the original comp three times and brought them into another comp. I've offset each layer a little on the Y axis so you can see them all:



So you can see that the wiggle() function is deterministic, and produces the same results for the same layer index across different comps. This is the problem you're experiencing. 

Fortunately there is a way of overriding the seed for the random number generator. That's the seedRandom() function. It's more commonly used for the random() and gaussRandom() functions, but it works for wiggle() and noise() too. The seedRandom() function takes two arguments, the first being a number, which seeds the random number generator, and the second a boolean which set whether the random number generator changes with time or if it stays the same. If you use the same seed the generator will always produce the same results, so usually on a layer you use the layer's index to set the seed. Since that won't work we can seed it with a unique number, derived from the comp's name. 

Here's the same comp with the three subcomps, only this time I've added the line

seedRandom(thisComp.name[thisComp.name.length -1], timeless =true);


before the wiggle() function. The term thisComp.name[thisComp.name.length -1] returns the last character of the comp name. And since my comps are called comp 1, comp 2 and comp 3 it returns the characters "1", "2", and "3". Since javascript is loosely typed, these characters get cast to ints for the seedRandom() first parameter, meaning that when I duplicate the comp I automatically get a unique random seed (I'd have to use regex or get the Ascii values of the letters or something if there were more than one digit, and it would break if I renamed the comps). Here's the final comp now 



Interestingly adding the seedRandom() for the first layer also effected any other layers in that comp. So if the first layer has a random seed based on the comp name, other layers using wiggle() will now get identical results. So overriding the random number generator is global within the scope of each composition (I just learned this). Here's the second comp, but with seedRandom() on the first layer (red), but not on the second layer(green) offset 1 pixel for visibility:



to avoid this gotcha, you can add back the index to the random seed:

seedRandom(index + thisComp.name[thisComp.name.length-1],timeless = true);


TL;DR

if your comps are named comp 1, comp 2, comp 3, etc., adding this line to any layers with a wiggle expression will generate unique wiggles:

seedRandom(index + thisComp.name[thisComp.name.length-1],timeless = true);