Open Peer Review is the practice of making the academic review process transparent by disclosing the names of all reviewers to the authors. Some publishers also make the review and author's response publicly available too as part of open peer review.
The Peer Reviewers Openness (PRO) Initiative aims to change the the cultural behaviour of scholarly publishing by requiring authors and reviewers to agree to open peer review wherever possible in order to promote transparency and openness.
For further information on the advantages and disadvantages of open peer review, the following articles may be of interest:
Ross-Hellauer T, Deppe A, Schmidt B (2017) Survey on open peer review: Attitudes and experience amongst editors, authors and reviewers. PLoS ONE 12(12): e0189311. https://doi.org/10.1371/journal.pone.0189311
Pöschl U (2012) Multi-stage open peer review: scientific evaluation integrating the strengths of traditional peer review with the virtues of transparency and self-regulation. Front. Comput. Neurosci. 6:33. doi: 10.3389/fncom.2012.00033
What happened when Elsevier tried open peer review? And which field says “no, thanks?” https://retractionwatch.com/2018/07/03/what-happened-when-elsevier-tried-open-peer-review-and-which-field-says-no-thanks/
Below are some publishers and specific journals who use Open Peer Review. This is an illustrative list rather than an exhaustive one.