Michael - thank you. I'm going to be "that guy" (sorry) and say that it's the refrain, not the verse. But you knew that. :-)
This kind of romantic/comic interplay is hardly uncommon, of course. Another Hart favorite that does the same kind of thing is "I Feel At Home With You" from A Connecticut Yankee. Later, Porter would do it with "Cherry Pies Out To Be You" from Out Of This World (the sweet upbeat version for Mercury and Chloe, followed by the comic fighting version for Niki and Juno). The most recent may be "Nothing Is Too Wonderful To Be True" from Dirty Rotten Scoundrels, where Christine's refrain sounds earnestly sweet, but Freddy's refrain is silly.
A longtime friend who used to be a theatre professor and director loved to point out the other facet of "I'll Tell The Man In The Street" - that it was Hart's way of debunking the typical romantic love song lyric cliches, particularly in the verse (even slyly referring to Kern and Hammerstein's "I've Told Every Little Star") in order to use more modern and less lofty images for love - telling the man on the street, putting it in the newspaper, broadcasting it on the radio, etc. A really clever lyric.