The very first footage screened for Paramount executives was of a poorly crafted puppet in front of a background of a badly drawn Eiffel Tower, prompting one executive in the audience to yell, "Oh God, they fucked us!" This was a prank pulled by the directors and the shot then pulls back to reveal a highly refined marionette manipulating the inferior one, then flies over beautifully detailed Parisian landscape full of believable yet cheesy marionettes. This actually ended up being the opening shot of the movie.
(at around 56 mins) The statue in Jong-Il Kim's palace is actually a human in heavy makeup. In the first scene in which he appears, you can see his eyes blinking.
George Clooney was a driving force in getting Matt Stone and Trey Parker's South Park (1997) to air. He also appeared in the show and the subsequent movie. As to their puppetry portrayal in this film, both he and Matt Damon are quoted as saying they would have been offended if they weren't in the film.
Paramount immediately greenlighted the idea of making a puppet movie, in the mistaken belief that it would be cheaper than a live action film.
(at around 1 min) As a subtle detail in the opening scene in Paris, the streets are paved with croissants.