You seem confused about what is returned from a function and the stuff that is printed to the screen/browser. That said, if I understand you correctly you want something like the following, leave the second() function as above.

if($tran eq 'edit'){
    second(common($val_1, $val_2));
}
else{
    first(common($val_1, $val_2));
}

sub common{
    my $arg_1 = shift;
    my $arg_2 = shift;
    my $val_1 = "Tested";
    my $val_2 = "Passed";

    return ($arg_1,$arg_2,$val_1,$val_1);
}

sub first
{
    my ($arg_1, $arg_2, $val_3, $val_4) = @_;

    print "I've got::: Val 1=$arg_1 and Val 2=$arg_2<br>\n";
    print "<br>And I have::: Val 3=$val_3 and Val 4=$val_4<br>\n";

    print "<br><a href=\"pass.pl?tran=edit\">Edit X</a>\n";
}
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