Conversations
The dashboard's /sms/ view groups every message — sent or received — by the other phone number, so a reply shows up next to what prompted it instead of in a separate inbox.
The contact list
Each row is one counterpart number, regardless of whether your last exchange with them was something you sent or something they sent you. A direction filter (All / Sent / Received) narrows the list without changing how it's grouped.
Inside a thread
Opening a number shows every message with it in chronological order, sent and received interleaved, with a direction indicator on each line. It's a read of exactly one account's data — even if a number has texted other Connecfy accounts too, nothing outside your own account ever appears.
What populates it
Nothing separate to configure. Every outbound send and, once receiving is turned on, every inbound report lands in this same view automatically — there's no independent "conversations" data source to keep in sync.