Customers only see Portal conversations
The customer portal only displays portal-channel threads. Your text and email conversations with them are invisible on the portal. Those live on the customer's own phone and email client.
"Customer sees you as" identity
The Messages tab shows what name the customer sees you as: your name, or "*Company* Admin" when privacy mode is on. Configurable in Settings → Customer Portal.
Online status
You can optionally show customers when staff are online in the portal (Settings → Customer Portal).
Unmatched conversations
Texts and emails from unknown senders are visible to admins only. Nothing is silently dropped, and nothing leaks to other customers.