If you want to write a recursive procedure, you might want the filter the common keys that point to hash refs, apply the procedure to them, and then use Corion's "simple" method.

use Array::Utils qw( intersect );
my @refkeys1 = grep { ref $ref1->{$_} } keys %$ref1;
my @refkeys2 = grep { ref $ref2->{$_} } keys %$ref2;
my @common = intersect( @refkeys1, @refkeys2 );
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UPDATE: Here is a complete example which I only tested on your data provided. As with all recursive procedures more testing would be needed...

use strict;
use warnings;
use Data::Dumper;
use Array::Utils qw( intersect );

my $ref1={
      ref1_specific_key => 'some value',
      common_key => {
             a => 1,
             common_subkey => 'foo'
            }
     };
    
my $ref2= {
       ref2_specific_key => 'some other value',
       common_key => {
              b => 2,
              common_subkey => 'bar'
             }
      };

sub mergehash {
  my ($ref1, $ref2) = @_;
  my @refkeys1 = grep { ref $ref1->{$_} } keys %$ref1;
  my @refkeys2 = grep { ref $ref2->{$_} } keys %$ref2;
  my @common = intersect( @refkeys1, @refkeys2 );
  for my $c (@common) {
    $ref2->{$c} = mergehash( $ref1->{$c}, $ref2->{$c} );
  }
  return { %$ref1, %$ref2 };
}

my $output = mergehash($ref1,$ref2);
print Dumper $output;
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