First off, Facebook impressed me with their data scientist recruiting process if only because they gave out study guides so candidates know exactly what they will be tested on. That said, I'll try not to repeat what is in those guides here and walk through the process.
The interview consists of a recruiter phone screen, a virtual interview with a data scientist, and on-site interviews. My recruiter found me through LinkedIn. The initial recruiter phone screen is pretty much the only time you get the "tell me about yourself" question. Since this recruiter came somewhat out-of-blue I scheduled my virtual interview for a month after this screen and studied with the given materials.
My virtual interview with a data scientist consisted of a SQL question and thinking through how I would solve a question (determining if a conversation was happening in the comments) algorithmically. Really enjoyable interview. No statistics/math was asked during this interview.
I moved forward to the on-sites. Of the 7 employees I spoke to, 5 had a PhD, which spooked me out a little bit as a 23-year old with just an undergrad but hey they brought me to Menlo Park so I must have some potential ¯\_(ツ)_/¯. The study guide lays out the content of the interviews: two are about thinking through product questions algorithmically (no code required, just sketch out thoughts), two are SQL whiteboarding, and one is statistics.
Prior to on-sites, I spent a lot of time looking at Facebook's product/news releases and writing responses to "hmm, how would I measure if this is working?" and this basically prepped me well for the Product interviews. I froze for a long time on the first SQL interview over a small point and this essentially scuttled my chances of getting the job; my advice would be to start white boarding and not worry too much about going back if you need to change code for an edge case -- I did this for the second SQL interview and did much better. I did alright on the applied math question. Just remember how to compute expectations for a probability distribution; my particular question ended up using the geometric distribution.
I didn't get the job but I think that's OK: this position is pretty self-guided (people who excel at this position ask great questions of the data and manage time smartly without too much oversight) and you are essentially equal to the PM in driving the direction of products. Not really what I'm looking for right now early in my career but something I'll definitely revisit later on.