Moving Media Files between Media Libraries

Apologies in advance if my question is unclear (I’m having difficulty locating an answer from various plugin developers).

I am trying to allow my WP users (buddypress) to move media files they access on the front end of our site (e.g., we publish a page that includes embedded media items from our WP media library) into their individual media libraries (e.g., I select an image file and am given the option to add to my media folder).

I am very new to web development, so I understand if I am lacking the language to express the action/event I hope to learn how to implement via WP.

Best,

Ben