Cryptocurrencies redefined how money can be stored and transferred among users. However, independent of the amount being sent, public blockchain-based cryptocurrencies suffer from high transaction waiting times and fees. These drawbacks hinder the wide use of cryptocurrencies by masses. To address these challenges, payment channel network concept is touted as the most viable solution to be used for micro-payments. The idea is exchanging the ownership of money by keeping the state of the accounts locally. The users inform the blockchain rarely, which decreases the load on the blockchain. Specifically, payment channel networks can provide transaction approvals in seconds by charging a nominal fee proportional to the payment amount. Such attraction on payment channel networks inspired many recent studies which focus on how to design them and allocate channels such that the transactions will be secure and efficient. However, as payment channel networks are emerging and reaching large number of users, privacy issues are becoming more relevant that raise concerns about exposing not only individual habits but also businesses' revenues. In this paper, we first propose a categorization of the existing payment networks formed on top of blockchain-backed cryptocurrencies. After discussing several emerging attacks on user/business privacy in these payment channel networks, we qualitatively evaluate them based on a number of privacy metrics that relate to our case. Based on the discussions on the strengths and weaknesses of the approaches, we offer possible directions for research for the future of privacy based payment channel networks.