Applied online and received a very prompt e-mail reply that was "boilerplate", containing a list of questions for me to answer. Answered the questions the same day and received an HR e-mail the next day asking for a block of time periods that I could have a telephone interview. My telephone interview was with a Senior Operations Manager and came one week after my initial application. Within 24 hours of that interview I received another HR e-mail inviting me to in-person "pod" interviews in three weeks. Three days later I received another e-mail inviting me to an earlier set of "pod" interviews, which I accepted. The pod interviews followed the script as described very well elsewhere on glassdoor. I got the flow problem, one I had seen verbatim on glassdoor. As other have said, getting the answer correct is important, but knowing the answer is only the beginning. I was given several change scenarios verbally so the interviewer could see how I reasoned on my feet. In my opinion, you have to be prepared for the flow problem just to get past first base, but you cannot really cram to beat the "What if" scenarios that follow. You either think analytically or you don't, and this data-driven company wants analytical leaders at a minimum.
I interviewed with three Senior Operations Managers in three separate one-on-one interviews. There were no trick or shock questions, just the "Tell me a time when..." or "How would you handle XXX?" In every case, I felt the interviewers went out of their way to introduce themeselves fully and tell their story first, and they always kept open some time at the end for my questions. I appreciated their candor in their answers and felt like they were really interested in me as an individual, rather than simply trying to weed me out. The HR rep running the pod told us very clearly when we would hear from them and what the three possible outcomes would be: 1) We get an offer, 2) we get an offer for a position we had not applied for (due to them finding some other strengths in areas such as facilities, finance, etc), 3) they will not be moving forward on our application and they would not be able to give ANY feedback. I received an offer 20 days after my initial application.
In general, the people I dealt with were professional and friendly. They were open about the fact that Amazon is growing very quickly and is looking for leaders who know how to develop the next geenration of leaders. "Prepare to work quite hard and be rewarded with a nice package" seemed to be the tone. On the other hand, most of my scheduled interactions were rescheduled once, and their e-mail communications were clear to expect that, since their operations leaders are quite busy. I tried to be positive and outgoing and felt like that was reciprocated by everyone I dealt with.