We saw some in Curacao living naturally in the lagoons there, it's free to see them. Unless you want to wade a few hundred meters into the lagoon (please don't they will probably freak out), you wouldn't get close to them. In Aruba they were brought to a private island and some of the hotels and it's a touristy day trip experience one pays quite a bit for,.
If you want to experience flamingos up close the best place we went was in Nassau, Bahamas where there is also a small private garden you can walk around together with flamingos. It was called something like Astrada gardens, very reasonably priced and there are parrots and nice plants there too.