They can manifest in any of the realms. Indeed they aren’t really “deities” as such, more like the living quality of compassion, wisdom, etc. For instance Tibetans think of the Dalai Lama as being Chenrezig (Same as Avelokiteshvara). This doesn’t mean he is the “only Chenrezig”. It just means that they think he carries some kind of connection or blessing of compassion.

Anyone who shows compassion, in that moment is manifesting Chenrezig. But the boundless compassion, wisdom and openness of the Buddha is something that for us who think of ourselves as limited beings, we can’t identify with easily. So it has something sacred and divine about it, and so to us, it’s something we see as external to the limited way we see the world.

Pure land schools talk about “Avelokiteshvara pure land” as if it is some other realm you can go to. But even then - it’s not like a heavenly realm of carefree happiness with nothing to do and only pure joy. It’s a realm where everything carries the message of the path to enlightenment and there is no falling back and you open out more and more to the message of compassion. It’s very much to do with your own understanding and it’s possible to see this very world as a pure land. A pure land not in the sense that everything is wonderful with pink fluffy clouds and nothing to think about. But in the sense that whatever happens, that it’s got the message of the Buddha, that it’s opening you to compassion, wisdom, patience also. It’s not easy to practice that way but it is possible. For Buddhists following the path of Shakyamuni Buddha then you can think of it that by bringing his teachings into the world that he has brought a blessing that let’s us relate to it as a kind of a pure land if we can but relate to the dharma and see this.

So, when the Tibetans think of everyone as manifesting the compassion of Chenrezig in their actions - that’s to do with this idea that the pure land unlimited compassion of Avelokiteshvara is something that’s never far from us, in any moment indeed. The Zen Buddhists have similar ideas. And Therevadhans also have the idea that enlightenment is something you can realize in any moment. So there isn’t really this distinction as if Avelokiteshvara is in some other realm. This vast open compassion is here right here, right now.