You have two people sitting at a table. A is sitting on one side and B is on the side to A's right. So you say 'B is sitting to the right of A' or 'on A's right hand' if you want to be archaic. However. You have two people sitting on the same side of a table. B is next to A. You can say 'B is next to A', fine, but you can also say 'B is sitting to the right of A' or 'on the right hand of A.'
Is there any wording that specifically suggests that A and B are sitting on different sides of the same table, and not on the same side?