Copernicus Publications publishes a host of journals, many on the behalf of non-profit scholarly and scientific societies [1]. For some or all of their journals, including those by the European Geosciences Union, peer review is open and there is a month-long public discussion forum for each manuscript.
I've got a paper in review with one of the journals right now, but several of my colleagues have expressed caution with open peer review. I think a lot of scientists are worried about making work public that hasn't been privately vetted during peer review, and maybe being caught with an embarrassing mistake.