It says in Choshen Mishpat Siman 9 that it is forbidden for a Dayan (judge) to take a shochad (bribe). What about if a particular Dayan is dealing with a case between Reuven and Shimon, both of whom want to give the Dayan $500 each. Is this still called a "shochad" if he is receiving the same amount of money from each side of the case?
R' Hershel Schachter was quoted in an interview with Ami Magazine, lamenting how Batei Din often (mal)function (emphasis mine):
Q: Do you have a problem with the borerim system [in which two of the dayanim are chosen by the litigants and the two dayanim choose a third]?
A: The borerim system is also a shanda. A lot of the borerim act like toanim. I was involved in a din Torah. The borer took shochad (bribes). I had to resign from the case. He felt insulted. It was before Rosh Hashanah, and he told me that he was not going to be mochel [forgive] me. I told him, “I don’t need mechila. You took shochad. You’re pasul to be a dayan.” It says in Shulchan Aruch that you can’t have one litigant pay his dayan and the other pay his dayan, unless, which Reb Moshe writes in a teshuva, it is clear that both are being paid the same amount, in which case each one can pay his dayan and they both pay the third. But that isn’t what happens. They don’t pay the same amount. The payment depends on how long each one bothers the dayan. So they don’t pay the same amount and it is true shochad.
See this post from Bein Din L'Din, explaining that it's generally prohibited.
The Sm'a in that siman s.k. 2 says explicitly that it is not allowed even though the regular logic of bribery does not apply. He says to look at what he wrote in the Drisha. There he quotes a gemara in Ksuvos 105a which says taking bribery even in order to judge justly is assur. Tosafos and the Meiri there disallow the taking of money from both parties in the form of bribery. Taking from both parties in the form of payment for judgement would be a different issue, as brought in the Shulchan Aruch in that siman siff 5.