Why and how commonly are scientists concerned about open peer review?
This answer discusses a survey that indicates blind reviews are much preferred to open ones.  The main reason, from that answer, seems to be


Many people don't want to review non-anonymously.  I know many people on both sides of the fence on this issue.  Once concern is that authors are often sensitive to criticism and may harbor ill-feelings against a person who gives an unfavorable review (especially if the reviewers have not properly understood the paper).  Consequently, open reviewers may be biased (more so than blind reviewers) to colleagues they respect or want a good relationship with.  


Fomite mentions a couple of other issues, but I see this as the main one (and it seems the survey does too).  Some other potential issues are


Certain "rivals" or "critics" can try to ruin your career by giving unfairly harsh reviews.
Some well-respect person could make a comment based on a cursory read of the paper that ends up heavily influencing the reviews, and maybe no one reads the paper carefully enough.