I modified your first program (and called it one.pl) to:

  #! /usr/bin/perl -w
  use strict;
  $|=1;

  print "Once\n";
  sleep 2;

  print "Input Please?\n";

  my $line = <STDIN>;
  chomp $line;
  print "You got: $line\n";

  sleep 5;
  print "Twice\n";
  exit 0;
[download]

Then I modified your second program to use IPC::Open2 to:

  #! /usr/bin/perl -w
  use strict;
  use IPC::Open2;

  my $reader;
  my $writer;
  my $pid = open2 ($reader, $writer, './one.pl');

  print scalar <$reader>;
  print scalar <$reader>;
  print $writer "Some input.\n";
  print scalar <$reader>;
  print scalar <$reader>;

  close $reader;
  close $writer;
  exit 0;
[download]

It now does something like what you expected. Now, I believe you really should use Expect to do this kind of thing, which gives you nice control over the I/O of the first program, and the ability to handle errors, which this code does not.