Running 4.8 (alias 4.08 in case you ever wondered about that), here is what I figured out about giving permissions to fully use the Documents module for both editing and uploading files.
1. Name a Security Role, like "content editor" as an example. It doesn't matter if it's a role or role under a group as it concerns assigning the security permission to a user. Naming a group is just for management convenience (example: Staff > content editor, moderator, etc..) Each role either independently or beneath a group can be assigned its own security setting.
2. Go to User Group and give permission to the user for that role.
3. Go to a Module Settings and sort filter by either the group or role.
4. Hashmark Edit for that group or role.
At this point, your user can edit, but has no access to files, except external links (also usable for HTML/Links Module).
5. Go to Admin > File Manager > allow "role" to use Documents folder. Bottom of page, filter by group or role. Allow this "role" to view and edit the folder.
This will allow your user to upload files in the Documents Module. From a management POV, it's important to trust your users for this last setting.
Val