Depending on your needs, it may be better to use the global variable $main::DEBUG directly rather than a stored local copy such as $self->{DEBUG}. The reason is you might want to localise $main::DEBUG.

That is, if package Deb prints reams of debugging information when $main::DEBUG is set, you might find it useful in debugging to say

package Deb;

sub frob {
  # do stuff...
  print_debugging_info if $main::DEBUG;
  # do more stuff...
}
# ...


package main;

$::DEBUG = undef;      # usually don't want debugging information

my $deb1 = new Deb (...);
my $deb2 = new Deb (...);

# do stuff with $deb1, $deb2, but no debugging information

{
  local $::DEBUG = 1;  # buggy bit

  $deb1->frob(42);
  $deb2->nitz(17);
}

# no more debugging output

# ...
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local is usually a very good match for various logs.