Carolyn, ... Good question. I think so. When I got back to JFK (1st entry) from Nassau last December, I think I just got off the plane. Our baggage was checked all the way home. No showing my passport. No customs declaration that I remember. But, I do remember wrong more often as I get older.
I do remember three, separate security checks at the Nassau airport - one by US; one by Bahamian officials; then, another one right as we boarded the plane. I also remember showing my passport at all three.
When my wife and I arrived in Nassau, we got in a LONG line to clear Bahamian immigration. Then, we picked up our checked luggage. Walked right by the Bahamian customs people with it. No questions at all.
Maybe somebody else can validate or correct my memory.
BTW, despite my protests, my wife bought 6 bottles of rum thinking she could carry them on the plane. She quickly found out she couldn't. But, we put all 6 in our checked luggage wrapped in dirty clothes and every single one got home with no leaks. I was shocked.
... eom