The easiest way to add records to a Zotero group is through the desktop client. If group members have synced their desktop client with their Zotero accounts, they will see their groups at the bottom of the left-hand column in the Zotero client.
Collections (folders) can be used to organize the group library, and are created by right-clicking on the group library folder. Subfolders can be added by right-clicking on a folder. Items can be dragged from a member's personal library and added to the group library folder or to a group library subfolder.
NOTE: This creates a duplicate of the item that now belongs to the group. If the member later updates or deletes the item from their personal library, those changes will not be reflected in the group library. Likewise, changes made to the new item in the group library will not be reflected on the original item in the member's library.