I think this will do what you need. It uses the recursive subpatterns described in the perlre documentation. By the time I finished getting my example working I saw that roboticus had already mentioned them. (I had never used the recursive regex method, so it was a good learning experience for me).

Edit:   Fixed some comments (specifically capture group numbering), and captured a little bit more.

Edit 2: Added output.

Edit 3: Allow keyword 'report' (somehow missed it the first time).

#!/usr/bin/perl
#
#  References:
#    http://perldoc.perl.org/perlre.html (See section on 'PARNO')
##

use strict;
use warnings;
use feature qw( say );
use Method::Signatures;


##################
## Main Program ##
##################
my $str = 'private function convert_wa_date_strings(iv_beg string, iv_
+end string, iv_read_date date, iv_step char(6)) returns (date, date, 
+char(1))';

recursive_function_parsing_regex($str);


#################
## Subroutines ##
#################
func recursive_function_parsing_regex($str) {
  my $re = qr{
      (                           # Paren group 1 -- full function
        (?:
          (private|public)        # Paren group 2 -- optional 'private
+' or 'public'
        \s+)?

        (function)                # Paren group 3 -- required 'functio
+n' keyword
        \s*                       # Optional space after 'function'
        (\w+)                     # Paren group 4 -- function name
        (                         # Paren group 5 -- args in parens
            \(
                (                 # Paren group 6 -- contents of paren
+s
                  (?:
                    (?> [^()]+ )  # Non-parens without backtracking
                  |
                    (?5)          # Recurse to start of paren group 5
                  )*
                )
            \)
        )
        (?:                       # Optional return value
            \s+
            returns\s*
            (                     # Paren group 7 -- return args in pa
+rens
              \(
                 (                 # Paren group 8 -- return args
                   (?:
                     (?> [^()]+ )  # Non-parens without backtracking
                   |
                     (?7)          # Recurse to start of paren group 7
                   )*
                 )
              \)
            )
        )?
      )
    }x;

    if ($str !~ /$re/) {
        say "No match for '$str'";
        return;
    }

    my ($full, $pp, $func, $name, $par, $args, $ret, $rargs) = ($1, $2
+ || "", $3, $4, $5, $6, $7 || "", $8 || "");

    say "Match!";
    say "  \$full  => '$full'";     # Full expression
    say "  \$pp    => '$pp'";       # Optional 'private' or 'public' k
+eyword
    say "  \$func  => '$func'";     # 'function' keyword
    say "  \$name  => '$name'";     # Function name
    say "  \$par   => '$par'";      # Func args (in parens)
    say "  \$args  => '$args'";     # Func args (no parens)
    say "  \$ret   => '$ret'";      # Optional return args (in parens)
    say "  \$rargs => '$rargs'";    # Optional return args (no parens)
}
[download]

Result:

Match!
  $full  => 'private function convert_wa_date_strings(iv_beg string, i
+v_end string, iv_read_date date, iv_step char(6)) returns (date, date
+, char(1))'
  $pp    => 'private'
  $func  => 'function'
  $name  => 'convert_wa_date_strings'
  $par   => '(iv_beg string, iv_end string, iv_read_date date, iv_step
+ char(6))'
  $args  => 'iv_beg string, iv_end string, iv_read_date date, iv_step 
+char(6)'
  $ret   => '(date, date, char(1))'
  $rargs => 'date, date, char(1)'
[download]