Why do monks have one shoulder exposed sometimes and both shoulders covered other times?
There is a very small rule about having the robes fully covered when we are in a populated areas (sekhiyā 3). Although it is a very small rule, it can say a great deal about the monk and his respect for the monastic code, especially in the afternoons or evenings.
"Populated areas" refers to when we are outside of the monastery and outside of the forest. Buddhist Shrines in a city like Shwedagon Pagoda can be considered a "monastery" and it is OK and sometimes culturally proper to go bare shouldered inside. However, some may choose to stay to be fully covered if there are many women around.
Monks often leave the monastery for collecting alms food. Because of that, whether one cares about the rules or not, it has become a tradition that is followed when one collects food more or less to show that "one is collecting food." At Pa-Auk, we go fully clothed at the alms food hall which is inside the monastery for the purpose of tradition. The alms-giving section where the women are allowed, is treated like an "outside and populated" place even though it is not required by the rules.
Contrary, if a monk is living in a dwelling place, even with a temporary determination as a residence for himself, he may make himself comfortable inside and switch to one shoulder. However, if it is a typical hotel, apartment building, or residential house, with women anywhere inside the building, the monk should not lie down at night in such a place according to a different rule (pācittiyā). For instance, in 2007, when Venerable Pa-Auk Sayadaw traveled to the India Holy Sites, he slept in a camper van to avoid all problems at hotels.
(most monasteries/temples in the world, likewise in India, are unallowable because they handle money, etc (nissaggiyā pācittiyā 10,18,19,20) ).
For monks that follow the rules strictly, if you see one on the streets, he will be fully covered during any time of the day. If he is travelling, seeing a doctor or even riding in a car from one location to the next, even inside the same village, he will take the time to wrap himself completely.
Just as a child who shows respect to his parents by following all of their rules, monks should reflect in the same way about The Buddha.