Persistent Identifiers (PIDs) and their infrastructures are vital for accountability, reproducibility, and credibility in research. They contribute to Open Science and Open Scholarship by fostering transparency, identification, and referencing of scientific outputs. PIDs like DOIs, ORCID iDs, and ROR iDs are essential for research funders, universities, and researchers to enhance visibility, reproducibility, and attribution. The success of PIDs depends on their added values, trustworthiness, and absence of risks. A comprehensive investigation conducted in 2022 explored the requirements for a well-functioning PID infrastructure. The study focused on risk and trust, adopting a stakeholder and case study approach involving PID Authorities, Service Providers, Managers, Owners, and Users. A literature review clarified trust definitions and trustee nature, along with risk dimensions and management strategies. Qualitative interviews with 15 international PID experts were conducted, transcribed, and coded. Findings were analyzed across technological, social, economic, and political dimensions. This contribution highlights PIDs as socio-technical systems and crucial infrastructures for Open Science. The study revealed organizational, political, and social risks, as well as substantial trust in PID infrastructures. Recommendations are provided for PID infrastructure development, and suggestions were given for academic institutions, funding organizations, libraries, and publishers to improve PIDs' permanence and reliability in line with the FAIR principles.