You can configure outgoing e-mail of SharePoint via SharePoint Central administration. SharePoint will use this outgoing e-mail server to send emails or notifications. Shetab Windows Live Authentication for SharePoint use same mechanism to send verification email. click "Configure outgoing e-mail settings" to see how to do it.
Problem
You can send emails to your local network but when you may not send email to another email server such as Yahoo or Gmail even if you setup all settings correctly, your emails may not received at destination. Today's the major email servers do not accept any email from unknown servers, they may mark mail as spam and it amy going to spam box or they may entirely reject the email. The email may even reach to destination but it will just by chance and very unstable.
Solution 1 – Use well known email hosting
You need to use a well-know email hosting to send and receive emails to another network.
Solution 2 – Use Gmail account
You can create use a Gmail account or Google app, and configure your windows email server to bypass outgoing message to it.
Learn How to Configure IIS SMTP Server to forward emails using a Gmail account?
Note: Gmail support default TCP port 25, so you don't need change it to port 527 as mentioned in the article.