I don't think you do - what I did to solve this problem was used a parallel region #pragma omp parallel shared(...) private(...) and allocated the array dynamically inside the parallel region. Try this:
#include <stdio.h>
#include <stdlib.h>
#include <malloc.h>
/* compile with gcc -o test2 -fopenmp test2.c */
int main(int argc, char** argv)
{
    int i = 0;
    int size = 20;
    int* a = (int*) calloc(size, sizeof(int));
    int* b = (int*) calloc(size, sizeof(int));
    int* c;
    for ( i = 0; i < size; i++ )
    {
        a[i] = i;
        b[i] = size-i;
        printf("[BEFORE] At %d: a=%d, b=%d\n", i, a[i], b[i]);
    }
    #pragma omp parallel shared(a,b) private(c,i)
    {
        c = (int*) calloc(3, sizeof(int));
        #pragma omp for
        for ( i = 0; i < size; i++ )
        {
            c[0] = 5*a[i];
            c[1] = 2*b[i];
            c[2] = -2*i;
            a[i] = c[0]+c[1]+c[2];
            c[0] = 4*a[i];
            c[1] = -1*b[i];
            c[2] = i;
            b[i] = c[0]+c[1]+c[2];
        }
        free(c);
    }
    for ( i = 0; i < size; i++ )
    {
        printf("[AFTER] At %d: a=%d, b=%d\n", i, a[i], b[i]);
    }
}
That to me produced the same results as my earlier experiment program:
#include <stdio.h>
#include <stdlib.h>
#include <malloc.h>
/* compile with gcc -o test1 -fopenmp test1.c */
int main(int argc, char** argv)
{
    int i = 0;
    int size = 20;
    int* a = (int*) calloc(size, sizeof(int));
    int* b = (int*) calloc(size, sizeof(int));
    for ( i = 0; i < size; i++ )
    {
        a[i] = i;
        b[i] = size-i;
        printf("[BEFORE] At %d: a=%d, b=%d\n", i, a[i], b[i]);
    }
    #pragma omp parallel for shared(a,b) private(i)
    for ( i = 0; i < size; i++ )
    {
        a[i] = 5*a[i]+2*b[i]-2*i;
        b[i] = 4*a[i]-b[i]+i;
    }
    for ( i = 0; i < size; i++ )
    {
        printf("[AFTER] At %d: a=%d, b=%d\n", i, a[i], b[i]);
    }
}
At a guess I'd say because OpenMP can't deduce the size of the array it can't be private - only compile-time arrays can be done this way. I get segfaults when I try to private a dynamically allocated array, presumably because of access violations. Allocating the array on each thread as if you'd written this using pthreads makes sense and solves the issue.