My desktop computer is not connected to the internet a the moment, but I want to test the notification icon in the taskbar.

Is it possible?

If Claws is sending email via an internet service, such as your ISP's mail server or gmail, no, because the mail would have to go to the internet and come back.

I just wondering wether an email could be sent locally or not.

Its not such a big problem though, I can wait untill I get online again.

It certainly can, if you are running your own mail server in your local network.

You may have a local mail server set up for the local computer--I know that Slackware and Debian install with that enabled. As user1 I can email user2 on the local computer.

But I've never looked into setting up a mail server to send email between computers.

Sendmail and postfix are the two most common mail transport agents (MTAs). Exim is a common command line client for such installation (that sounds like I know something about this--I really don't, except there's a lot I don't know about how to do this).

This article will tell you more: http://flurdy.com/docs/postfix/

I know I've gotten emails on my Slackware computers from my Slackware computers, but I've never looked into sending one.