Being unfamiliar with the name pillow moss I did a google search. It would appear what is being sold as pillow moss or frog moss is Leucobryum glaucum, also known as pincushion moss. If this is the same species, I would guess that what is being sold was harvested form the wild. This is a slow-growing species that occurs primarily on wood humus, which is essentially completely decomposed wood. I've propagated it myself before using divided 'cushions' and keeping them in a partially covered terrarium. The rate of spread was quite slow, maybe doubling size after many months. Like many other mosses a mist irrigation system would probably improve growth.

I am not sure on the scientific name. Over here we call it pillow moss. It grows naturally, in large quantities.

I did not buy this moss, I harvested it from a valley. It grows in abundance in valleys, also on shady spots on roof tops. I've also seen it grown immensely on bus shelters!

To be honest I've never really seen it grow on wood. 90% of the time I've seen it grown on soil, concrete, sandstone and limestone and other stone types.

How would one cultivate this? Just plant it in some soil and water it regularly?

Would the buttermilk/yogurt method work on this species?

It is not Leucobryum as that genus does not occur in Malta or its neighboring islands.
It is apparent from what you are saying is that the moss grows well where water runs off rapidly and the surface does not get mucky wet.
So you need to note the amount of moisture the mosses get where they grow. Check weather information for your area. My guess is that there is substantial rain and high humidity in the areas where the moss is found, either naturally or by human irrigation.
to grow I would try clean sand from a fresh water stream and use either collected rain water or buy distilled water. It would probably be best to try and grow in a shallow plastic pot or container in which many holes have been punched in the bottom so that the water will drain out. Cover the bottom of the container with a piece of window screen to keep the sand from washing out.
Keep the moss in a shady and the coolest spot you have.
If finding success then you can experiment with more exposure to sun and varying other conditions as well.
Good luck.