We have created an insert notification, is there any way to process the document in the subscriber without publishing it?

When you create an Insert Notification or any other, It by default creates a Publishable document associated with the notification and when a DB change Happens, then the activity will be notified by the corresponding Notification as a Publishable document to Broker. This is an auto Publish.

Just create a trigger with this document type and a subscribing service; you would not need to write a publish step in the Subscribing service again.