Private conversations (or private messages) are organized similarly to threads. We use the name private conversation to differentiate from private messages, which may not be thread-based. Thus, sent and received messages both show up in your "Private Conversations" list in their respective conversation.
Yes, there is a red indicator that appears in the upper right area (Private conversation inbox) when a new private message is sent to you along with email notification. Private messages are saved in that same area until you choose to delete them. Why are you asking?