• You should put spaces around your Binding Operators. Regular expressions already look confusing to a lot of people. Same for before the parentheses of your for loop, but that's less of an issue.
• One might naively expect that the hash in question contains the counts, and you correct that right before output. That could be confusing if the two loops were more complex or separated by a lot of code. Rather than fixing it at the end, you could initialize the value to 1:

  while($text=~/\b(\w+)\b(?=.*\b\1\b)/g){
      $words{$1} //= 1;  # initialize empty value to one
      $words{$1}++;
  }
  [download]

  See Logical Defined Or and Assignment Operators for details on what I did there.
• I usually name my hashes based upon how the key is related to the value. In this case, for example, %hits could be natural because when you look at it on the screen, you see the number of hits you got -- $hits{$word} reads to me as hits for word in English. On the other hand, %words sounds like I'm going to get words back, not a number. With that first change, it might be %count, %repetitions or %reps. Perl more than most programming languages is supposed to read following common grammar.

use strict;
use warnings;
use 5.10.0;

my %reps;
my $text = "and him him lad has him done and john has has";

while($text =~ /\b(\w+)\b(?=.*\b\1\b)/g){
    $reps{$1} //= 1;
    $reps{$1}++;
}

foreach my $key (keys %words){
    say "$key: $reps{$key}";
}
[download]