Siddiqui, who was nominated in the best performance by an actor category for his Netflix movie ''Serious Men'', lost out to British star David Tennant.
While Das' Netflix comedy special ''Vir Das: For India'' lost out to ''Call My Agent'' Season 4 from France in the comedy category.
The comedian, who has been in news back home for his ''I Come From Two Indias'' monologue, said he loved the French drama.
''I was nominated for best comedy at the International Emmy Awards, for jokes. Call My Agent, a gigantic beautiful show I love won. But I got this medal, and ate this fantastic salad with a very interesting crispy cheese topping. It was an honour to represent my country. Thanks so much to the @iemmys For India. It’s always for India. #VirDasForIndia,'' he wrote on Instagram.
Das has received praise as well as police complaints for his monologue in Washigton recently.
Sen's ''Aarya'', a Disney+ Hotstar series directed by Ram Madhvani, was nominated in the best drama series category alongside Chile's ''El Presidente'', Israel's ''Tehran'' and the second season of ''There She Goes'' from the UK.
Israel's ''Tehran'' topped the drama series category, ending India's run at the yearly awards.
Madhvani on Monday had hoped to bring the trophy home. “Here we are after taking a 16-hour flight, we are hoping when we go back, we will win but even if we don’t it is a big big thing to be nominated,'' he said on Monday.
Last year, director Richie Mehta's ''Delhi Crime'', fronted by Shefali Shah, won the best drama series honours at the International Emmys.
Organised by the International Academy of Television Arts & Sciences, the International Emmys aim at ''recognizing excellence in television produced outside of the U.S. for over 48 years''.