One of the rules for monks and nuns is that inside a monastery or when in the forest, the right shoulder should remain uncovered. When in inhabited areas, both shoulders should be covered. We can see both of these styles of wearing the robes shown in Buddha images.
One shoulder uncovered (All images):
Both shoulders covered (All images):
In many Buddha statues from the Mahayana and Tibetan tradition it is common to see the left shoulder covered completely and the right shoulder covered partially: