The main problem is, these people are not here in the US. Hence their language skills can not get any better just by trying to talk to someone on the phone. Most of them are in India, using a VoIP number provided by the only one or two person(s) who are actually here to collect commissions.
Since I am not actively seeking new employment, I can be selective about answering my phone. Hence, I never have to talk to them. The ones, who are desperate, start sending emails. At that point, if I anything catches my attention, I usually copy the first couple of sentences from the technical portion of job requirements and perform a google search. If I am unlucky, I will see the same job description posted by tons of "consulting" companies (read as commission based head hunters from India) and I really have to sift through the chaff to get the real wheat. Most of the time though, you will find in the first page of results, which company is actually hiring. If you are really interested, you should go around the so called consulting companies and directly deal with the employer.
Sometimes, employer is not the actual company but an outsourcing firm, usually from India. Likes of Wipro, Tata, etc. In which case I immediately lose interest, knowing how these sweatshops operate. If you end up with a prominent consulting company, based in US, such as Robert Half, Experis/Manpower, KPMG etc, you can contact these companies. Usually, the search result you find, will have some sort of initial contact information.
To warn you about the worst part of this scam, these people make you accept something to the sense of they are the sole submitter of your name for the position. If you agree to that, you will be giving them a portion of your paycheck, without them doing nothing but finding the job listing you can easily find and finding your resume at the same time. In most of these cases, they are making unsolicited submissions. And if you accepted it (and to be submitted to these positions you have to accept that offer), it puts the actual hiring company into a bad situation. They can not extend you an offer easily. There are ways around these scam artists, but if the company has enough talent to pick and choose from, your candidacy will not be considered to avoid such hassle.
So, all in all, talking to these people with thick accents is mostly a losing battle from the start. But if you are going to do it anyway, my advice is, never give them an expected salary or hourly rate and never accept the first offer they are going to make. If you do, you are handing them undeserved amount of money from your hard work.
I know I went a bit off tangent here but having to deal with these people for the good part of last two decades,gave me a different perspective and thought you might benefit from that. If not, my apologies.