According to Reb Yehuda b'shmai Reb Eliezer, how physically did the goral
work? The sgan kohen gadol and the kohen gadol would both pick the goral with the rght hand so who's would go to the sier (goat) on the right and who's would be placed on the left? Since the sgan stands on the right of the kohen gadol would his go to the animal on the right?
The S'fas Emes explains that even according to the opinion that holds the Segan would pick one of the lots, the Segan did not place it on the head of the goat. The verse clearly states (Vayikra 16:8) that it was Aharon who would put the lots on the two goats. Therefore, after the Segan picks his lot with his right hand, he puts it into the left hand of the Kohen Gadol. The Kohen Gadol then places his left hand on the left goat and his right on the goat on his right.