I would never give my students one another's paper to grade. I would only pass their papers out face down and I would never announce a grade aloud. I feel that if they want their grades broadcasted, let them do it. I have seen on TV shows, etc. where teachers call out the student's name and then announce their grade (Saved by the Bell was notorious for that!

). Sometimes, my students don't discuss their grades. I have heard students lie to their peers (when they thought I wasn't listening) and say they had a better grade than they really earned.
The problem is that too much emphasis is placed on grades these days. Of course, grades are important, but I've seen students come to tears over getting an 89 instead of a 90. I know that one point can make a difference between an 'A' or a 'B,' but is it really that serious? Grades are not a measure of knowledge anymore these days. It's more of a competition and it all ties to self-esteem. Students feel that if they are "smart" and the "dumbest girl/boy in class" gets a better grade, then they are a COMPLETE failure. No, it may just mean that the "dumb kid" is better at memorization! I have weaker students in my class who have once or twice scored higher than some of my stronger kids. I have also overhead the stronger ones question the weak students about whether or not they cheated, etc.
We have to let our students know that grades are important, but they should not be the entire basis on which they value their self-worth.