That would only catch messages sent to the target, not messages that the target sent itself. For example if I send a iMessage to a friend on my Mac, I won't see my sent message on my iPhone later.
iMessage syncs the sent messages across the devices as well. The might be some conditions on the sync (e.g. messages sent before the device was added to the account, or message older than x months during which the device was offline, etc..) but I'm looking at my mac app and I definitely my iPhone messages.