Distributed ledger and blockchain systems are expected to make financial systems easier to audit, reduce counter-party risk and transfer assets seamlessly. The key concept is a token controlled by a cryptographic private key for spending, and represented by a public key for receiving and audit purposes. Ownership transfers are authorized with digital signatures and recorded on a ledger visible to numerous participants. Several ways to enhance the privacy of such ledgers have been proposed. In this paper we study two major techniques to enhance privacy of token transfers with the help of improved cryptography: M\"obius and CryptoNote. The comparison is illuminating: both techniques use "ring signatures" and some form of "stealth addressing" or key derivation techniques, yet each does it in a completely different way. M\"obius is more recent and operates in a more co-operative way (with permission) and is not yet specified at a sufficiently detailed level. Our primary goal is to explore the suitability of these two techniques for improving the privacy of payments on cryptographic ledgers. We explain various conflicting requirements and strategic choices which arise when trying to conceal the identity of participants and the exact details of transactions in our context while simultaneously enabling fast final settlement of tokens with a reasonable level of liquidity. We show that in these systems, third-party observers see obfuscated settlement. We finish with a summary of explicit warnings and advice for implementors of such systems.