Trust in the quality of scholarly publishing is vital for the integrity of science and medicine in society. An ongoing debate about peer review has divided its participants into those who defend pre-publication peer review (e.g., preprints), traditional peer review (TPR), and post-publication peer review (PPPR). Even though the quality of peer review can vary widely within and between journals, peer reviewers within TPR may be rewarded uniformly by reward schemes such as Clarivate’s Web of Science. Despite this, academics and the public are usually unaware of the content of peer reports or quality of peer review because the process is deemed confidential, except for rare cases of open peer review (OPR). In this paper, we argue that the confidentiality associated with TPR creates an opaque smoke-screen that prevents the direct and across-the-board scrutiny of peer reviewers or editorial handling, and may be a contributing factor to the current crisis of mistrust in science and academic publishing. We make two related arguments. Firstly, that peer reviewers within TPR whose reports’ content and quality cannot be independently or publicly verified or contested should not be rewarded by any peer reviewer rewards schemes, but should be in the case of OPR. Secondly, journals should not flout their “peer-reviewed” status unless they can provide publicly verifiable evidence of their peer review process, as in OPR. In other words, there should be “no reward without responsibility” for peer reviewers, as well as editors, journals and publishers claiming that the process or their journal is peer-reviewed, absent evidence.