1) SMS or MMS; no notification of text entry, nothing happens until you press "send". The system is store-and-forward, not instant.
You can request notification of delivery, usually deep in a configuration menu on your phone. This is a network service, so the operator would know, as would you. But hardly anyone uses this.
2) BBM; is a private, IP-based instant messaging system. Messaging is online, and some features like notification of entry are available. The owner of the BES (BlackBerry Enterprise Server) will know; the owner of a BIS (BB Internet Server, the hosted version) may know depending on the terms of agreement between them and RIM.
3) Apple iMessage, WhatsApp, etc: IP-based, usually XMPP messaging. Online and feature-rich. The operator wouldn't know but the owner of the XMPP server might.
Confusion may arise because iMessage provides an integrated view of the SMS and iMessage inboxes, choosing to use iMessage to anyone whose iMessage or other XMPP IM address it knows.