Can students run similarity checks on their own?

Yes, if the Student Precheck option is enabled in assignment settings.

Please refer to the Student Precheck section of the Student’s Guide for more information on how students can use this feature. 

Instructor can set the number of precheck attempts. Prechecked drafts (original submissions and similarity reports) will not be visible to the instructor, while students will have access to these files via the Drafts folder on the assignment page.

Note: In Student Precheck mode, draft submissions are checked against the Internet only.