Any user with permission to edit pages in a space can move attachments in that space to any page in Confluence.

Eg: suppose we have a user named StandardUser who has permission to edit pages in GeneralSpace, but no permission to view or edit RestrictedSpace, which contains a page predictably named Home.
StandardUser:

• goes to the attachments view of a page with attachments in GeneralSpace.
• clicks edit.
• types "RestrictedSpace:Home" into the Page field and clicks save.

The attachment is moved.

The user should really need the following permissions:
View Space for RestrictedSpace
Create Attachment for RestrictedSpace
Furthermore, the user should not be restricted from viewing or editing the target page by any page level restrictions.