Hi,
with Social Network plugin you already have a button to send message on others profiles, you can also send messages at your friend from message section (see screenshot)

Without Social Network plugin you can use Joomla contact integration, if user have a Joomla contact linked to some profile you will see a button on profile page to send email, this button redirect to contact page. Joomla have a plugin to generate contact when user register his account, this plugin is disabled by default. To enable you can go on Extensions->Plugin Manager and search for "User - Contact Creator"