This is just a simple nested foreach loop. outer loop iterates over all id's. inner loop iterates over all id's except the current id (skips itself which may or may not be part of the requirements?). Note: grep is used in a scalar context which is the number of matches.

#!/usr/bin/perl -w
use strict;

my %hash= ( 'ID1' => {
                   'myStr'         =>     'hello',
                   'description' => 'Description 1',
                   'yourStr'     => [
                                    'goodbye',
                                    'where'
                                    ]
                 },
          'ID2' => {
                    'myStr' => 'good',
                    'description' => 'Description 2',
                   'yourStr'     => [
                                       'hello',
                                    ]
                  },
          'ID3' => {
                    'myStr' => 'testvar',
                    'description' => 'Description 2',
                   'yourStr'     => [
                                    'hello',
                                    'good'
                                    ]
                  },                  
           );
           
foreach my $id (keys %hash) 
{
    my $mystr = $hash{$id}{myStr};
    foreach my $compareid (grep{$_ ne $id}keys %hash)
    {
        if (grep{ $_ eq $mystr} @{$hash{$compareid}{yourStr}})
        {
           print "$id myStr=$mystr found in $compareid yourStr=@{$hash
+{$compareid}{yourStr}}\n";
        }   
    }
}

__END__
ID1 myStr=hello found in ID3 yourStr=hello good
ID1 myStr=hello found in ID2 yourStr=hello
ID2 myStr=good found in ID3 yourStr=hello good
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