From what I've read, the Citizenship Act of India specifies that any citizen of India who obtains citizenship of any other country immediately ceases to be a citizen of India. At that point, it is his or her duty to surrender their passport to the nearest Indian mission abroad.

I guess that one could ignore this and keep on using their Indian passport - but doing so would be fraud and criminal misrepresentation.

My Indian passport says:

    WARNING
    If holder acquires Nationality 
    of another country, he should 
    Surrender This Passport 
    forthwith to the nearest Indian 
    Mission/Post Abroad. Un-
    authorized possession of this 
    Document shall Constitute an 
    Offence Punishable under 
    the Indian Passport Act 1967

>>Not sure about India

Not had this experience to comment on this first hand.

But as an Indian, I guess they ask you to surrender your Passport at the Indian embassy, and then ask you to apply for a PIO(Person of Indian origin) card.

How does the Indian government know that a person has another citizenship?

One way is when we enter India at borders, they check out exit dates in system. Usually people who are still indians would have US's some kind of visa. US Citizens will not be able to show any US visa.

There are heavy fines for not self surrendering the indian passport after US Citizenship within 6 months.

I think one way is taxes. But I guess there will be some sharing of databases.

On the other hand, I think, putting your US citizenship on risk for maintaining a illegal status would not make sense for a lot of people.