You are able to private message most people within Moodle, whether that be a fellow student or your tutor.  

Please follow the steps below.

1) Click on the intended student/tutor's name or avatar. This will take you to their profile.

2) Once in their profile, click on the message icon next to their name. This will take you into the messages section of the website.

You might notice this speech bubble icon in the top right of any page on Moodle. This enables you to access Moodle Messages service at any point when browsing the Moodle platform.