The way it is defined in the paper "Tweakable Block Ciphers" by Moses Liskov, Ronald L. Rivest and David Wagner:
A tweakable block cipher should also be secure, meaning that even if an
adversary has control of the tweak input, we want the tweakable block cipher
to remain secure.
For this operator, we call the new (second) input a tweak" rather than a nonce" or initialization vector," but the intent is similar.
so it is intended to be similar to an IV or nonce, not to be similar to a secret key.
Although a higher level mode of operation could still define the tweak to be secret, it certainly isn't intended to be, and it is unclear what the advantage would be, given that the block cipher itself doesn't depend on it.
So the only reason why the tweak would need to be kept secret is that it itself contains information that needs to be kept secret for other uses.