As an American, I could also say "The bank is at the right." This would imply at the right side of something, but it's assumed that you mean the street. So if you're giving directions, you could say, "Go ahead two blocks, turn left, and the bank is at the right."
Could this be used in a similar way in Dutch?
It depends on the context, and "on" is much more common in American English. I would say, "The soap is on the right side of the sink." The only example I can think of right now where you would use "at" is when giving directions.
Edit: I take that back. You can also use "at" when identifying the location of an object, if you are implying that the object is somewhere generally within the location. "My car is at the repair shop." "The books that I ordered are still at the bookstore." "I am waiting at the intersection of 1st Avenue and Main Street." My sense is that "on" is used when positioning an object relative to another object, while "at" is used when positioning a smaller thing within a bigger place.
Because it is ON the right side. Not BY the right side. Though in this particular sentence I am doubting whether they mean a bank as in a couch (piece of furniture) or a 'real' bank (financial institution).
I assume they mean a bank as in a couch and then the most 'logical' or fitting word would be indeed ON (Dutch: AAN). The couch is standing ON the right side. De bank staat aan de rechterkant.
'BY the right side (de bank staat bij de rechterkant) would not be even fully wrong I think (people would understand it) but it does sound a bit strange somehow in this case.