About message history on companion devices
On the multi-device beta, right after you link a companion device, your phone sends a copy of your most recent message history end-to-end encrypted to your newly linked device where it’s stored locally. Once sent from your phone, your encrypted message history is only stored on WhatsApp’s servers until it’s delivered to your companion device unless deleted from WhatsApp's servers earlier. Learn more about end-to-end encryption in this article.
Note: WhatsApp Desktop syncs more message history than WhatsApp Web.