Department stores in the U.S. (At least in Minneapolis) used to ferry payment via baskets on wires from the clerk up to an accounting room where the money was counted, change made, and delivered back to the customer via the same contraption.
I tried to implement a system like this once, with the goal being to reduce the amount of equipment and clutter on the service counter, but ran into issues of trust. I learned very quickly that if you remove the cash from sight of the customer (even by turning around or having a cash register under the counter) you invite potential fraud from the cashier or the customer. The cashier is able to skim, or the customer to claim they gave a different amount if the money disappears, even briefly. I gave up on it and kept the standard "customer-facing register" setup.
Various other places has used pneumatic tube systems.
That's a pretty widespread system. I even saw an antique one in a newsagent in Broken Hill (way outback Australia) a few years ago. They did a little demo for me when I asked what it was, though obviously it was only a curio these days.
Interesting, do you have a source so I could learn more?