OpenMP: are local variables automatically private?

Yes, local variables are automatically private.

The reason for the existence of the private clause is so that you don’t have to change your code.

The only way to parallelize the following code without the private clause

int i,j;
#pragma omp parallel for private(j)
for(i = 0; i < n; i++) {
    for(j = 0; j < n; j++) {
        //do something
    }
}

is to change the code. For example like this:

int i
#pragma omp parallel for
for(i = 0; i < n; i++) {
    int j;
    for(j = 0; j < n; j++) {
        //do something
    }
}

That’s perfectly valid C89/C90 code but one of the goals of OpenMP is not have to change your code except to add pragma statements which can be enabled or disabled at compile time.