Two steps is the magic:

use strict;
use warnings;

while (<DATA>) {
    chomp;
    print "$_: ";
    s/"(.*)"/$1/g;
    s/""/"/g;
    print ">$_<\n";
}

__DATA__
""
""""
"a"""""
"a""b""c""d""f"
[download]

Prints:

"": ><
"""": >"<
"a""""": >a""<
"a""b""c""d""f": >a"b"c"d"f<
[download]

Note that this fails if there is more than one quoted string in the string being processed.