opendir(DIR,$str);
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See opendir. That's for reading a directory.

If you want to write files into a directory, name it. open doesn't guess, and doesn't magically prepend a path component to the file name; opendir doesn't set a directory context for open.

for my $key (keys %hash){
  my $outputfile = "output_$key";
  $outputfile =~ s{/}{-}g;
  $outputfile = join '/', $str,$outputfile; # <---- this is missing
  print "Writing to $outputfile\n";
  open OUT,'>',$outputfile 
       or die "Could not open $outputfile : $!";
  print OUT $hash{$key};
  close OUT;
}
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Better alternative to join, because it is portable (Windows etc):

use File::Spec;
...
for my $key (keys %hash){
  ...
  $outputfile = File::Spec->catfile($str,$outputfile);
  ...
}
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update:

Re-reading the question and code, it looks like you mistook opendir for chdir which sets the current working directory. Cwd comes in handy:

#!/usr/bin/perl
use strict;
use warnings;
use Cwd;

my %hash=();

# input
# my $inputfile = 'nv.txt';
# open IN, '<',$inputfile 
#     or die "Could not open $inputfile : $!"; 

# we use the text after __DATA__ for input
while (<DATA>) {
  my ($key, $val) = split (/:/,$_,2);
  $hash{$key} .= $val;
}

# output
my $dir = "dict";
-d $dir or mkdir $dir;
my $cwd = getcwd(); # from Cwd imported above
chdir $dir; # go into directory 'dict'
for my $key (keys %hash){
  my $outputfile = "output_$key";
  $outputfile =~ s{/}{-}g;
  print "Writing to $outputfile\n";
  open OUT,'>',$outputfile 
       or die "Could not open $outputfile : $!";
  print OUT $hash{$key};
  close OUT;
}
chdir $cwd; # change directory back




__DATA__
test/right/case1:12: //comment
test/right/case1:344: //comment
test/wrong/case3:123:  //comment
Output expected:
test/right/case1: 12: //comment
                :344: // comment
test/wrong/case3:123: //comment
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