From IO::Socket::INET docs:

If "Listen" is defined then a listen socket is created, else if the socket type, which is derived from the protocol, is SOCK_STREAM then connect() is called.

You're using a connect() socket only, so your process family won't talk much. You need at least one process listening and the other connecting so they can speak; but please note that it will be a client/server (i.e. child/parent) dialog, so the parent won't be able to say a thing until the child requests. If you need a different kind of dialog, then you should change your path to the IPC route.

Update: This is the classic ping-pong example, that you could find helpful...

#!/usr/bin/perl

use IO::Socket::INET;

use strict;
use warnings;

if (my $pid = fork()) {
    my $listen = IO::Socket::INET->new(
        LocalAddr => 'localhost',
        LocalPort => 9000,
        Proto    => 'tcp',
        Listen   => 1,
    ) or die "Cannot create listening socket: $!\n";

    my $client = $listen->accept()
        or die "Cannot get a connection: $!\n";

    $client->autoflush(1);

    while (defined(my $line = <$client>)) {
        print "server < $line";
        sleep 1;
        pong($client);
    }
}
else {
    print "waiting for parent to setup...\n";
    sleep 1;

    my $sock = IO::Socket::INET->new(
        PeerAddr => 'localhost',
        PeerPort => 9000,
        Proto    => 'tcp',
        Type     => SOCK_STREAM
    ) or die "Cannot create connecting socket: $!\n";

    $sock->autoflush(1);

    ping($sock);
    while (defined(my $line = <$sock>)) {
        print "client < $line";
        sleep 1;
        ping($sock);
    }
}

sub ping {
    my ($fh) = @_;

    print "client > ping\n";
    print {$fh} "ping\n";
}

sub pong {
    my ($fh) = @_;

    print "server > pong\n";
    print {$fh} "pong\n";
}
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