Since the emergence of coronavirus disease–2019 (COVID-19) outbreak, every country has implemented digital solutions in the form of mobile applications, web-based frameworks, and/or integrated platforms in which huge amounts of personal data are collected for various purposes (e.g., contact tracing, suspect search, and quarantine monitoring). These systems not only collect basic data about individuals but, in most cases, very sensitive data like their movements, spatio-temporal activities, travel history, visits to churches/clubs, purchases, and social interactions. While collection and utilization of person-specific data in different contexts is essential to limiting the spread of COVID-19, it increases the chances of privacy breaches and personal data misuse. Recently, many privacy protection techniques (PPTs) have been proposed based on the person-specific data included in different data types (e.g., tables, graphs, matrixes, barcodes, and geospatial data), and epidemic containment strategies (ECSs) (contact tracing, quarantine monitoring, symptom reports, etc.) in order to minimize privacy breaches and to permit only the intended uses of such personal data. In this paper, we present an extensive review of the PPTs that have been recently proposed to address the diverse privacy requirements/concerns stemming from the COVID-19 pandemic. We describe the heterogeneous types of data collected to control this pandemic, and the corresponding PPTs, as well as the paradigm shifts in personal data handling brought on by this pandemic. We systemically map the recently proposed PPTs into various ECSs and data lifecycle phases, and present an in-depth review of existing PPTs and evaluation metrics employed for analysis of their suitability. We describe various PPTs developed during the COVID-19 period that leverage emerging technologies, such as federated learning, blockchain, privacy by design, and swarm learning, to name a few. Furthermore, we discuss the challenges of preserving individual privacy during a pandemic, the role of privacy regulations/laws, and promising future research directions. With this article, our aim is to highlight the recent PPTs that have been specifically proposed for the COVID-19 arena, and point out research gaps for future developments in this regard.