Jinja Kentei was started three years ago, and is run by the Foundation for Promoting Japanese Culture, which is extremely closely associated with Association of Shinto Shrines. (I suspect that there are legal reasons, to do with the regulations for religious corporations and secular foundations, for having two organisations, but I don’t know.) The Association of Shinto Shrines is overseeing the examination, with the goal of spreading accurate knowledge about Shinto and jinja.
The examination for Jinja Kentei is, of course, held only in Japanese, and is based on Japanese texts. Each examination is 100 multiple-choice questions, taken over the course of a little less than two hours. The examination now has three levels: Level 3, the easiest, Level 2, and Level 1, the hardest. Level 3 was first held in 2012, Level 2 in 2013, and Level 1 this year. In order to take Level 1, you must first pass Level 2, but you may take Level 2 without passing Level 3, and may take Levels 2 and 3 in the same year.
The fact that the examination and its texts are only available in Japanese is further evidence of a fact about the current situation: no-one can really get to know and understand Shinto unless they speak and read Japanese well. This is why my first recommendation for people with a serious interest in Shinto is that they start learning Japanese. It will take a long time to get to the necessary level, so you should start learning Japanese first, and look further into what material is available in English once you have that in progress.
Incidentally, if you read Japanese well, the official textbooks for the examination (six volumes so far) make up an extremely good introduction to Shinto. The most basic volume covers the real basics, then there is a volume of history, a book about the legends, a book about various matsuri held across Japan, a book about sengū, the ceremonies in which shrines are rebuilt or repaired, and the kami moved to the new structure, with a particular emphasis on Jingū at Ise. The Level 1 textbook goes into quite a lot of academic detail about important concepts, people, and organisations in Shinto, both now and historically, and is a good way to get more depth. The plan is to issue another Level 1 textbook next year, and there may be a new textbook for Level 2, as well, as Level 3 got a new text this year.
To return to the examination itself, I took each exam in the first year it was offered, and passed all of them. In total, 164 people passed Level 1 this year, a pass rate of 27.1%. Because the foundation publishes the distribution of the scores, giving the number of people with each total score, I also know that 145 of the people who passed did so with a lower score than I achieved. (12 had a higher score, with 3 people on the highest score of 90; there were a total of 7 people with my score of 85.)
In other words, not only do I know a lot about Shinto, I know more about it than almost all Japanese people, including almost all of those who have an interest in it. According to an article in Jinja Shinpō, the newspaper of the Association of Shinto Shrines, Level 3 is roughly equivalent to the information gained in the first year of the four-year course for training Shinto priests. (The practical skills are, of course, not covered in the exam.) I don’t know where Level 1 matches up to, however.