JavaScript (ES6), 100 90 110 102 96 bytes

a=>b=>a.map(v=>t[v],a.map((_,k)=>a.map((x,i)=>t[x]=t[y=b[i]]=Math.max(k?t[x]:x,k?t[y]:y)),t={}))

My initial solution was 90 bytes:

a=>b=>a.map(v=>t[v],a.map(_=>a.map((x,i)=>t[x]=t[y=b[i]]=Math.max(t[x]||x,t[y]||y)),t={}))

Although it's passing all provided test cases, it fails for something such as:

A = [0, -1], B = [-1, -1]

Test cases

let f =

a=>b=>a.map(v=>t[v],a.map((_,k)=>a.map((x,i)=>t[x]=t[y=b[i]]=Math.max(k?t[x]:x,k?t[y]:y)),t={}))

console.log(JSON.stringify(f([0])([0]))); // -> [0]
console.log(JSON.stringify(f([1])([2]))); // -> [2]
console.log(JSON.stringify(f([0,1,0])([2,1,0]))); // -> [2,1,2]
console.log(JSON.stringify(f([1,2,3])([0,0,1]))); // -> [3,3,3]
console.log(JSON.stringify(f([0,1,0,2])([-1,1,2,2]))); // -> [2,1,2,2]
console.log(JSON.stringify(f([1,0,1,-4])([-3,-1,-2,2]))); // -> [1,0,1,2]
console.log(JSON.stringify(f([1,2,3,-2])([1,0,-3,-2]))); // -> [1,2,3,-2]
console.log(JSON.stringify(f([-3,-2,-1,0,1])([-1,-1,-1,-1,-1]))); // -> [1,1,1,1,1]
console.log(JSON.stringify(f([-3,-2,-1,0,1])([2,-1,0,1,-3]))); // -> [2,2,2,2,2]
console.log(JSON.stringify(f([-3,5,5,3,1])([4,2,3,1,2]))); // -> [4,5,5,5,5]
console.log(JSON.stringify(f([4,0,2,-5,0])([0,4,-5,3,5]))); // -> [5,5,3,3,5]
console.log(JSON.stringify(f([-2,4,-2,3,2,4,1,1])([-2,4,1,2,2,3,1,-2]))); // -> [1,4,1,4,4,4,1,1]
console.log(JSON.stringify(f([-10,-20,-11,12,-18,14,-8,-1,-14,15,-17,18,18,-6,3,1,15,-15,-19,-19])([-13,6,-4,3,19,1,-10,-15,-15,11,6,9,-11,18,6,6,-5,-15,7,-11]))); // -> [-8,14,18,14,19,14,-8,-1,-1,15,14,18,18,18,14,14,15,-1,18,18]
console.log(JSON.stringify(f([20,15,2,4,-10,-4,-19,15,-5,2,13,-3,-18,-5,-6,0,3,-6,3,-17])([-18,7,6,19,-8,-4,-16,-1,13,-18,8,8,-16,17,-9,14,-2,-12,7,6]))); // -> [20,15,20,19,-8,-4,20,15,17,20,17,17,20,17,-6,14,15,-6,15,20]
console.log(JSON.stringify(f([0,-1])([-1,-1]))); // -> [0,0]

CJam, 27 bytes

l~_])\z_,*f{{_2$&,*|}/:e>}p

Try it online! Test suite.

Explanation

l~       e# Read and evaluate input, dumping arrays A and B on the stack.
_        e# Copy B.
])\      e# Wrap in array, pull off B, swap. Gives B [A B] on the stack.
z        e# Transpose the [A B] matrix to get a list of all equivalent pairs.
_,*      e# Repeat this list by the number of pairs. This is to ensure that the
         e# following procedure is applied often enough to allow transitive
         e# equivalences to propagate.
f{       e# Map this block over B, passing in the list of pairs each time...
  {      e#   For each pair...
    _2$  e#     Copy both the pair and the current value/list.
    &,   e#     Get the length of their intersection. If this is non-zero,
         e#     the current pair belongs to the current equivalence class.
    *    e#     Repeat the pair that many times.
    |    e#     Set union between the current value/list and the repeated pair.
         e#     This adds the pair to the current list iff that list already
         e#     contains one value from the pair.
  }/
  :e>    e#   Get the maximum value of this equivalence class.
}
p        e# Pretty print.

Python 2, 91 bytes

f=lambda a,b:[a<x>b.update(b&set(x)and x)and b or max(f(zip(a,b)*len(a),{x})[0])for x in a]

Python, 86 bytes

f=lambda a,b:a*(a==b)or f(*[map({x:y for x,y in zip(a,b)if x<y}.get,x,x)for x in b,a])

Simultaneously updates both lists by replacing each value in the first list by the corresponding element in the second list if it's greater. The replacement is done with map on a dictionary's get method. Then, swaps the lists, and repeats until they are equal.

Php, 266 241 213 200 bytes

Solution:

function u($x,$y){foreach($x as$i=>$j){$k[$y[$i]][]=$j;$k[$j][]=$y[$i];}$h=function($c,&$w)use($k,&$h){$w[]=$c;foreach($k[$c]as$u)!in_array($u,$w)&&$h($u,$w);return max($w);};return array_map($h,$x);}

Usage: u([1,2,3], [0,0,1]); returns the desired array.

Not-so golfed:

function unify($x, $y)
{
    foreach($x as $i=>$j) {
        $k[$y[$i]][] = $j;
        $k[$j][] = $y[$i];
    }

    $h = function ($c, &$w=[]) use ($k, &$h) {
        $w[] = $c;
        foreach($k[$c] as $u)
            !in_array($u, $w) && $h($u, $w);
        return max($w);
    };

    return array_map($h, $x);
}

Pyth, 13 bytes

eMumS{s@#dGGC

Try it online: Demonstration

Explanation:

Start with each pair. Iteratively extend each pair (list) with overlapping lists, deduplicate the elements and sort. Stop once this process converges. Print the maximum of each list.

Mathematica, 56 bytes

#/.($|##->Max@##&@@@ConnectedComponents@Thread[#<->#2])&

Dyalog APL, 29 28 bytes

⌈/¨({∪¨,/∘.{⍵/⍨≢⍺∩⍵}⍨⍵}⍣≡,¨)

Same idea as the Pyth solution.

Java, 273 263 bytes

interface Z{int z(int x);default Z g(int m,int n){return x->{for(int t;x!=(t=x==m?z(n):z(x));)x=t;return x;};}static void f(int[]a,int[]b){Z y=x->x;int i=0,v;for(int u:a){u=y.z(u);v=y.z(b[i++]);if(u<v)y=y.g(u,v);if(v<u)y=y.g(v,u);}i=0;for(int u:a)a[i++]=y.z(u);}}

The method f(int[]a,int[]b) solves the challenge.

interface Z{

  //should return an "equivalent" integer
  int z(int x);

  //return a Z lambda where the 1st arg is "equivalent" to the 2nd arg
  default Z g(int m,int n){
    return x->{
      for(int t;x!=(t=x==m?z(n):z(x));) //always find the last equivalent number for x
        x=t;
      return x;
    };
  }

  //solve the challenge
  static void f(int[]a,int[]b){
    Z y=x->x; //start off with all numbers only equivalent only to themselves
    int i=0,v;
    for(int u:a){
      u=y.z(u); //get a's element's equivalent number
      v=y.z(b[i++]); //get b's element's equivalent number
      if(u<v)y=y.g(u,v); //if a's was smaller than b's, make a's equivalent to b's
      if(v<u)y=y.g(v,u); //if b's was smaller than a's, make b's equivalent to a's
    }
    i=0;
    for(int u:a) //overwrite a with each element's equivalent value
      a[i++]=y.z(u);
  }

}

First go through both arrays and keep track of equivalent numbers. Then modify every element in the first array to have the equivalent numbers stored.

Python, 522 bytes

a = [-2,4,-2,3,2,4,1,1]
b = [-2,4,1,2,2,3,1,-2]
m = {}
visited = {}
for i in range(len(a)):
    if a[i] in m:
        if b[i] not in m[a[i]]:
            m[a[i]].append(b[i])
    else:
        l = []
        l.append(b[i])
        m[a[i]] = l
    if b[i] in m:
        if a[i] not in m[b[i]]:
            m[b[i]].append(a[i])
    else:
        l = []
        l.append(a[i])
        m[b[i]] = l

def dfs(v, maximum):
    if v > maximum:
        maximum = v
    visited[v] = True
    for n in m[v]:
        if not visited[n]:
            d = dfs(n, maximum)
            if d > v:
                maximum = d
    return maximum

result = []
for k in range(len(a)):
    for q in m:
        visited[q] = False
    result.append(max(dfs(a[k], a[k]), dfs(b[k], b[k])))

print(result)

Explanation

Make a table of values corresponding to each unique element in in both arrays (a and b in this case). For example if

a = [0,1,0] 
b = [2,1,0]

then the table would be:

0:[0,2]
1:[1]
2:[0]

then apply depth first search, so, for example, assume I pick the leftmost element in a the value is then 0 and 0 has the equivalences: 0 and 2. Since 0 has already been visited, go to 2. 2 has the equivalences: 0. So the best result for choosing the leftmost element in a is 2. Here's the tree:

   0   
 /   \
0     2
       \
        0

and you want to take the largest value in there, so the result is 2.