There are no FAA requirements for becoming a flight test engineer (FTE); however, EASA has a formal certification process. This also means that you can become an FTE with no flying training at all.
There are several well-traveled roads to becoming a Flight Test professional. I illustrate nine broad paths here. For specifics on becoming an FTE in the military, including attendance at a Test Pilot School (which is not required), I wrote this.
Many start their flight test career at the widest part of the funnel. That is to say that, normally an organization will hire engineers and provide them with the training and development they need to become FTEs. This is good way to enter the pipeline.
There is a global organization of FTEs known as the Society of Flight Test Engineers (www.sfte.org), and it includes people who have become an FTE in every possible way. Its members can answer general questions, like the one posed above, and more specific questions that may apply at particular milestones along one's personal journey. SFTE has also formally responded to EASA's requirement for formal certification here.