Zero-knowledge Watermark Detection (ZKWMD) is a promising and powerful means to improve the security of digital watermarking schemes in the context of various copyright-protection applications: in contrast to standard watermarking schemes, ZKWMD allows a proving party to prove to untrusted parties the presence of hidden information (embedded in digital data) without requiring to disclose this information or any other secrets necessary to detect its presence. However, typical applications presume the embedded information to have certain properties such as to be drawn from a specific probability distribution, and/or to have a specific form to counter ambiguity attacks. Hence, additional verifications must be performed that are more involved since the input to a ZKWMD protocol is cryptographically concealed. We present concrete and practical protocols to securely perform these verifications as complementary protocols to ZKWMD. In this context we consider two different approaches whose deployment depends on the underlying applications: the first one allows to securely prove that the concealed information (watermark) suffices certain desired properties, whereas the second approach allows both parties to jointly, securely and verifiably generate this information with the desired properties.