The NOTIFY without an XML body you received just after a subscription is correct. rfc3265 says:
This SUBSCRIBE request will be confirmed with a final response. 200-class responses indicate that the subscription has been accepted, and that a NOTIFY will be sent immediately.
You can disable such notifies via send_fast_notify presence module parameter.
What about further notifications. Kamailio handles subscriptions and sends notifications when it receives PUBLISH requests. Using terms of the rfc3856 kamailio is a "presence server" and you need something to act as "Presence User Agent".
You can take a look at pua_dialoginfo module. I haven't used it, but its documentation says it can send PUBLISH requests based on active dialogs from the "dialog" module.