Ask first, and most people will not mind. That is the whole of the rule for photographing people. If you don't share a language, a gesture at the camera and a questioning look does the job — it is understood everywhere. In monasteries, most can be photographed and taking pictures outdoors is generally fine; indoors, anywhere photography isn't allowed has a clear sign up.
Photographing people: ask, then shoot
The etiquette is not complicated and it is not local to anywhere in particular — it is the same courtesy you would extend at home. Ask before you raise the camera. The difference here is that the answer is very often yes, and people are frequently happy to be photographed.
If you can't say it in words, say it without them. Lift the camera slightly, raise your eyebrows, tilt your head — a question. Nobody needs a translation to read that. In my experience it works, and the exchange usually ends better than a photograph taken from behind a long lens would have.
Monasteries: outside is easy, inside read the signs
Most monasteries can be photographed, and photographing outdoors — the halls, the courtyards, the architecture, the prayer wheels, the people crossing the square — is generally not a problem at all.
Inside is where you have to pay attention, and the good news is that you don't have to guess: where photography is not permitted, there is a sign, usually right at the door. Where there is no sign, the default on the routes I drive is that photographing the halls and the outside of things is normal.
Two habits cover almost everything. Keep the camera off the statues in any case, and ask before photographing anyone who is praying or in the middle of practice. That is not me inventing a rule — it is simply what asking first looks like when the person is busy with something that matters to them.
The wider point for photographers
This part of the world is unusually generous to a camera. The light is hard and clean up here, the landscape is enormous, and the people are, in the main, unbothered by being photographed — which is not true everywhere, and is worth not ruining.
Which is the real argument for asking. It costs a few seconds, it almost always gets you the picture, and it keeps the next photographer's ask easy too.
Questions I get on this one
Do I need permission to photograph people?
Can I photograph inside monasteries?
What about monks and nuns?
What if someone says no?
Written from photographing and driving these places rather than from a guidebook. Local custom varies from monastery to monastery — when in doubt, ask the person in front of you. Related: is China safe · every route I've written up · all essentials