As Amazon.com Inc. and Netflix Inc. look to build up in India amid a global expansion, they are following two diverging scripts.

On trips to Mumbai this year, Amazon Studios chief Roy Price sipped tea with Indian actor Amitabh Bachchan, visited the mansion of the country’s heartthrob Shah Rukh Khan and watched “Bollywood” musical numbers on set, as he scouted for locally appealing programming for Amazon’s forthcoming video service.

Though...