Via a reference from a colleague, I met an Amazon employee for an "informational interview". (My friend knew there were openings in the department but didn't know exactly what they were.) That was a great conversation. We talked about the specific domain and my background, then turned to the openings and narrowed down to one that was the best fit. It sounded like they were in no hurry and would wait to hire only exactly the right people for the team. Then he brought out a sheet of standard old-fashioned behavioral interview questions ("Describe a time when you had to make a difficult decision") and I sweated through it, but all the time we were also talking about the domain and the various things Amazon was doing, most of which I hadn't been aware of, which made me very interested in pursuing the position. It ended with him asking for my resume, and outlining the next steps of phone screens and then the on-site interview.
A week later, after prodding my contact, I was contacted by a recruiter, who set up a phone interview. First was with the actual hiring manager--but he didn't want to talk at all about the domain of the job (my area of expertise and the reason I was interested), he wanted to talk about abstracted or hypothetical Product Management situations. Rather, he wanted me to invent situations and then analyze them--kind of like having to write your own test questions, and then answer them! I recognize that that demonstrates cognitive ability, but I'm not sure what it has to do with the actual job of product management. I thought I did OK, at least survived.
Then they set up a second call, which I took as a sign of continued interest. This was with a different manager, not actually in the same department. His questions were also abstract and hypothetical, but he was much better about prompting me through the process and was a lot easier to talk to. I felt like I aced it.
Then, much to my surprise, I got an email from the recruiter saying that they were "pursuing other candidates". And that due to company policy no feedback on my interviews would be available, thank you and goodbye!