On Paradise there was no captain's table per se, but the captain can invite you to dine in the officer's mess which is actually very good.
There is also an invitations-only captain's dinner for certain VIP guests. We went to this one on my first paradise cruise. There are several tables with various officers. One for instance had the captain and chief engineer, another had the hotel director and staff captain.
On other ships I have heard it is luck of the draw really, this is secondhand but i've heard that on some ships/lines they have social hosts etc. on the first day that look for prospective diners
Holiday 5-day Western Caribbean
Liberty 8 day Western