When I use the Ambassador, are other players supposed to be able to see if I swap cards or not?

No. You can choose to swap 0, 1 or 2 of your cards (if you have that many), but other players shouldn't know which option you choose. The best method is to mix up all the cards in your hand and then choose which one(s) to keep. All the other players should be able to see is that you return 2 cards to the deck.
Related Rule(s)