I often think Text::Balanced is going to be the answer, but after I implement a solution using Text::Balanced I'm rarely satisfied that it is the least complex solution:

use strict;
use warnings;
use Text::Balanced 'extract_bracketed';

my $string = q/the users contain (bbc (333)) BLAH BLAH (ddc (223)) BLA
+H BLAH(ccc (123))/;


while( length $string ) {
    my( $name, $id, $rest ) = extract_record($string);
    print "Name: $name, ID: $id\n";
    $string = $rest;
}

sub extract_record {
    my $string = shift;
    my( $unwanted, $wanted, $rest ) = extract($string);
    return (extract($wanted))[0,1], $rest;
}

sub extract {
    my $input = shift;
    my( $balanced, $rest, $prefix ) = extract_bracketed($input, '()', 
+qr/[^(]*/);
    $balanced =~ s/^\(|\)$//g if length $balanced;
    return  map { trim($_) } $prefix, $balanced, $rest;
}


sub trim {
    my $string = shift;
    $string =~ s/^\s+|\s+$//g;
    return $string;
}
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Regexp::Common has Regexp::Common::balanced which would probably facilitate a reasonably robust and possibly less complicated solution (left as an exercise for the reader, as I've lost interest at this point. ;)

Update: But a pure-regexp approach would work for data that is relatively simple, as the OP's data seems to be:

my $string = q/the users contain (bbc (333)) BLAH BLAH (ddc (223)) BLA
+H BLAH(ccc (123))/;

my $re = qr/
  \(  (?<name>  [^(]+  )  # Capture name.
  \(  (?<id>    [^)]+  )  # Capture ID.
  \)\)         # Closing parens.
/x;

print "Name: $+{name}, ID: $+{id}\n" while $string =~ m/$re/g;
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...which is pretty much back to what you started with. ;)