How do I send email to another machine on my local network without hitting the internet?

Email - at least email using standard email programs such as you describe, assumes and requires that there be a mail server. The very thing you clearly state you don't have.

There are several reasons for this, ranging from how email is funneled into the correct accounts, to the very protocols that are used for transferring email from machine to machine.

A good example is that your email program knows how to send email using a protocol called "SMTP" - Simple Mail Transport Protocol. In fact, that's how email is sent from server to server as well, until it reaches the final server that hosts the email account for which the message is destined for. That server collects the mail destined for that account, and waits until it's contacted by your email program.

When downloading email your program uses either POP3 (Post Office Protocol version 3) or in some, cases IMAP (Internet Message Access Protocol) to request that the mail server deliver the messages associated with a specific account.

SMTP and POP3 (or IMAP) are very different protocols, and while it's not obvious on the surface, they actually serve very different needs. SMTP, in a sense, says "here's a message, go send it on to its recipient", while POP3 says things like "I'd like message numbers 2, 5 and 24 for the account named 'leo' with the password 'password'".

So to even further grossly oversimplify the problem, a mail program sends using SMTP, and therefore has no way to transfer a message directly to a program that, of course, uses POP3 or IMAP to receive.

There simply must be a mail server for what you and I consider normal email.

Now, it is possible to set up your own mail server on your LAN. You could even choose to make one of the two machines you mention a mail server. But I will warn you that setting up and running a mail server is not a job for the squeamish. Mail servers are complicated beasts, and getting them set up and connected properly can have you quickly pulling your hair out. I know, because I've done it. Repeatedly.

I used to run a mail server on my LAN - Merak Mail Server for Windows. If you're at all technically inclined, that may be a reasonable approach. Under Linux there are many, many mail server options, which require more detailed technical knowledge. (I'm sure readers familiar with the issue will also make suggestions in the comments to this article.)

But the short answer for most "mere mortals" is that in order to send mail from one machine to another, even on your local network, you will need a mail server involved somewhere. If that's not something you want to rely on, then transferring information with more traditional file sharing might be more appropriate, depending on what you're attempting to do.