Which Christian branches/sects have information about their belief system hidden from new members but revealed to established ones?
Historically this has been done in the early days of Christianity. The Early Church practiced this method, in some areas, in order to avoid ridicule, scorn, misunderstanding and possible blasphemy from the part of pagans.
In parts of the Early Christian Church, many aspects of Christian theology, including some sacraments and sacramentals, the so-called disciplina arcani, were kept hidden from the pagans lest they become objects of ridicule, and were also introduced gradually to catechumens or new converts. As the Age of Persecution ended, the secrecy was gradually relaxed. But the term continued to be used, and the same word is used in the Eastern Orthodox churches to describe "mysteries" and "sacraments". This is not usually so in the West, although theologically many aspects of sacraments are recognized as mysteries in the main sense described above, especially (for those churches accepting it) the doctrine of transubstantiation in the Eucharist. Hence Pope Paul VI's papal encyclical of 3 September 1965 on the Eucharist was titled, from its opening words, Mysterium fidei. In the Roman Rite Catholic Mass within or immediately after the formula of consecration of the wine, the celebrant says "The mystery of faith". Originally the term "Mystery" was used for the sacraments generally in both the East and the West, as shown from the "Mystagogical Homilies" of St. Cyril of Jerusalem and the work, On the Mysteries by St. Ambrose of Milan.
Although all the official doctrines of Christian churches have long been fully public, the loosely defined area of Christian thought called Christian mysticism often concerns the contemplation of sacred mysteries and may include the development of personal theories about them, undertaken in the knowledge that they can never be fully apprehended by man. - Sacred mysteries (Wikipedia)
The Knights of Columbus within the Catholic Church had have secret initiation ceremonies, but these do not engage on Church beliefs. They are simply keeping their ceremonies secret, thus equating secret to private! In January 2020, the Knights decided to drop their secret initiation ceremonies.(See: In major change, Knights of Columbus drop secret initiation ceremony)
”Secret” is not always the best phrase to employ at all times in reference to certain situations within Christianity. For example the the term secret for Vatican’s Secret Archives (Library)! A better understanding for the word for secretum would be private.
The Vatican said on Monday the new name would be the Vatican Apostolic Archives. This removes any potentially “negative nuances” from the Latin word “secretum”, which the pope said in a decree was closer to “private” or “reserved” than “secret” when the archives were first named in about 1610.
They have not been secret in practice for a long time anyway: like most state archives, they are open to qualified researchers after a period of time. The collection of papers, documents and parchments dates as far back as the eighth century, making the archives one of the world’s most important research centres. - Vatican's Secret Archives no longer officially secret after renaming