If you speak with an accent in English, you may notice that some people tend to understand you better than others.  It's intuitive that those with more experience with accents would be better at understanding them, but researchers Janet van Hell (Penn State) and Sarah Grey (Fordham University) looked into how the brain responds differently to errors in speech when the speaker has a foreign accent versus native speakers.  Their results showed that inexperienced listeners actually performed similarly in their ability to identify errors overall in American accented vs Chinese-English accented speech, but that there was a measurable difference in comprehension of grammatical errors depending on whether the listener was able to identify the type of accent they heard or not.  Those who correctly identified the speaker as having a Chinese-English accent were better able to identify grammatical errors than those who were unable to specify the accent they heard. 

What does this mean for non-native speakers of English?  One could interpret this information as further motivation to work towards improving their American accent to deal with these difficulties. In addition, it might be advisable to try to remain patient with inexperienced listeners and recognize that with time and exposure to accented speech, they will improve in their ability to overcome these limitations on their comprehension.

Source:
Penn State. (2017, April 20). Recognizing foreign accents helps brains process accented speech. ScienceDaily. Retrieved April 27, 2017 from www.sciencedaily.com/releases/2017/04/170420141739.htm

While having an accent does not always interfere with being understood when speaking English, it can affect how your listener perceives what you are saying.  The goal of American Accent training is not to eliminate an accent, but to eliminate the problems which may arise when your accent interferes with being understood.  Accent training can provide you with the tools you need to reduce communication issues and improve others' perception of your skills and abilities.
You can read about bias and accents in a recent article in Personnel Today.

Does the way you speak matter when making a business deal?  According to researchers at the University of Cape Town in South Africa, yes, it does.  They examined issues of race and accent by experimenting with a methodology called trust games, which uses a combination of game theory, economics and psychology.  What they discovered is that the subjects were most likely to place their trust (in the form of a financial investment) in those who looked and sounded the same as themselves.  The researchers drew the conclusion that inherent biases are at play when it comes to accent, perceived status, and trust.  You can read more about this study here.