You've been told about the problems with your usage of push and chomp, but I thought you might be interested in seeing a more "Perlish" way of doing this.

#!/usr/bin/perl

use strict;
use warnings;

print 'Enter a list of numbers separated by spaces: ';
chomp(my $input = <STDIN>);
my @arr1 = split /\s+/, $input;

my $count = @arr1;
print "Enter another $count numbers separated by spaces: ";
chomp($input = <STDIN>);

my @arr2 = split /\s+/, $input;

unless (@arr1 == @arr2) {
  die "ERROR: Arrays are different lengths\n";
}

my @sums = map { $arr1[$_] + $arr2[$_] } 0 .. $#arr1;

print "@sums\n";
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