For a scheme to be at least CPA secure, one has to randomize the scheme. In case of a secret key encryption scheme, the used randomness will typically become part of the cipher text as in the case of an IV and is "public" (This is not a necessary requirement but how most algorithms work).
Now, if I switch to public key encryption schemes, the randomness will typically not be "public". The decryption algorithm simply removes it without necessarily really knowing it, like for example in the case of LWE or McEliece.
Now my question is: Is this a necessary requirement for public key encryption schemes that at least some of the randomness is not explicitly given as part of the cipher text?
Moreover, might it even be the case that all randomness that effects security has to be non-public? To be more precise, does all randomness that I cannot replace with a public constant value, without harming the security of the scheme have to be private.
(As a motivating example of randomness that can not effect security: I can always add randomness by first XORing the plaintext with a random bit string and adding this bit string to the cipher text but the scheme would be as secure as if I XORed the all zero string instead. Hence, this randomness could be omitted without harming the security.)