I am an owner of a SP (online) Site within my company. We would like to give all users the ability to modify file/folder permissions under the Document Libraries they currently have access to. (e.g. be able to make a folder private so only they have access to it.) Where precisely are the settings to manage this option?
When you create any folder, just click on the ellipsus besides folder (3 dots) -> shared with -> advanced settings. This is the page where you will find permissions on that particular folder.
By default it will be inherited from parent (library in this case). Click on "stop inheriting from parent". And now you can add/edit/remove permissions for that particular folder. Give the person who wants to manage access to the folder a minimum "Edit" level permission on the folder. Or just give "full control" to make someone the owner of that folder.
For SharePoint Online, go library settings and then find the link "Permission for this document library". Then you can stop inheriting and create unique permission for this library. Add all users who need to manage permission level to a certain group and then grant the group with enough permission.
Note: It is always not recommended to give too many users the role to manage permission.