Interesting set of answers, which I see mainly come from North America and again demonstrate a difference between American English and British English.
For us (British English) the verb "to haggle" is informal (in terms of the language) and is used mainly in situations which are considered less formal (in real life). So for example it's usual to haggle at a street market, where prices might not be obvious, but less normal in a store, where prices are clearly stated (and often fixed). There's no sense of the conversation being emotional, adversarial or negative. It's just completely normal in the right situation. I think this might relate to the culture of each individual country - I now live in Italy and I note that people here are much more likely to expect some sort of discount than is normal in the UK. So I guess it's reasonable to assume that the way people think about haggling depends on the culture, and therefore it's possible that the word is considered differently in other English speaking countries and communities (such as India, South Africa, Australia,etc)?
On the other hand we don't very often use 'bargain' as a verb, and when we do it's not usually to describe a transaction between a customer and a shopkeeper or market stall owner. It's used mainly in situations where groups that represent large numbers of people try to reach an agreement with some sort of authority. Trade unions negotiating with the government is the most obvious example that I can think of. But also perhaps a group of protesters trying to persuade the police to allow their demonstration to proceed. So to a certain extent that seems a bit more similar to some of the N American English replies