Does the BytePhase n8n integration work on both n8n Cloud and self-hosted n8n?

Yes. The HTTP Request node with a Header Auth credential works on either. The BytePhase community node needs n8n 1.x, and on n8n Cloud community nodes must be enabled. Every call goes out from n8n to BytePhase, so a self-hosted n8n on a private network works as long as it can reach the internet. Nothing needs to be exposed.