Within the Orthodox tradition, is there a minhag for the correct side of the mechitzah men and women should sit? Right side of the mechitzah for men, left for women; or left side of the mechitzah for men, and right side for women? Does it matter?
I don't think that there is any specific rule regarding the mechitzah placement. The Bet Hamikdash (Temple in Jerusalem) had a women's "gallery" which was like a balcony. From my understanding, most "European" fashioned shuls (synagogues) had a women's balcony. In NYC as well as many other U.S. cities where large shul buildings were constructed in the early 1900's, even prior to the mass immigration following the Holocaust, almost every shul had a women's balcony. I can't think of any shul from that era that was NOT built with a balcony.
Lincoln Square Synagogue, a.k.a. Rabbi Riskin's shul, is circular and the seats are arranged as in a stadium. I understand that at the time that it was built, there was much controversy regarding this arrangement - both b/c of people facing different directions when davening as well as the mechitzah problem b/c with different levels of seating, the mechitzah didn't serve the purpose of men not seeing women (and vice versa) while davening. Considering that many shidduchim eventually arose out of the Lincoln Square shul, perhaps, that was the idea in the first place :-)
There is no unified custom across Orthodox shuls.
As i commented above:
Not everyone has the mechitzah down the middle like that. I've noticed in many Sfardi souls, the women's section is in the back. Some places have a gallery on the second floor. Still others might have the men's section in the middle, and the women's split across the sides.
I have seen it in all those ways. From here, we can conclude that it depends on space constraints, how the shul is set up, and the mood of whoever put it together. :)