Deployment to Testflight is handled by Circle CI when you send a commit to the beta branch of Eigen, run make deploy locally to trigger it. There is a blog post on the process here.
Note that only one beta can be deployed at a time; teams should coordinate on how to use that availability.
There are two types of betas on Testflight: Internal and External.
Internal betas are the ones triggered by shipping code to the beta branch (or by running make ipa; make distribute locally.) These need no human intervention in order to ship them to a phone, usually just the processing time which comes to about 10-15 minutes. They go to anyone who is a user inside our iTunes Connect team. That tends to just be the product team.
External betas requires a quick review from Apple. So far these have taken about a day. Nothing too prohibiting, but as of yet, there is no automation with respect to shipping a beta to external developers. You need to go into iTunes Connect, click through to Eigen, then choose an internal beta to be exposed to the outer world.