You want to configure a mail-server for your Local Area Network. You do not have domain name.
It is a very basic requirement and it can be done easily. As you have already selected postfix, so no need to tell you about installation of mail server.
You have asked in your first post that you want to see local mails in mail client.
Install Squirrelmail for this.
If you have configured your system correctly, you can send mails to other IP's of same network as well as receive too.
All you can do using mail client Squirrelmail
What you have to do.
Setup DNS server and add the Mail server MX records in the forward and reverse zone files.
You need to add hostname entries in /etc/hosts file.
Configure your main.cf
myhostname = server.test.local
mydomain = test.local
myorigin = $mydomain
inet_interfaces = all
inet_protocols = all
mydestination = $myhostname, localhost.$mydomain, localhost, $mydomain
mynetworks = 192.168.1.0/24, 127.0.0.0/8
home_mailbox = Maildir/
Now install Dovecot
Dovecot is an open source IMAP and POP3 mail server for Unix/Linux systems.
Configure it correctly.
Now install and configure Squirrelmail.
Here is complete mail system for your local area network.