I was referred by a senior employee to the Amazon recruiters in India. In spite of that, I got a call about the same opportunity from a placement agency. I get spam calls everyday from such firms, so I said no. The very next day, the amazon recruiter called me, and then sent me the JD. Based on that, he arranged for a phone screen. The phone screen occured within a week.
Unfortunately, it was a completely negative interview, and it was evident that the interviewer was simply going through the motions. It lasted about an hour, where we went over my resume, my work done at previous firms and some design, data structure questions. [ This makes me wonder what would have happened if I had agreed to send my resume via the agency. ] I got the rejection mail within 24 hours.