Osmosis is simply the transfer of liquids through materials. Fish, birds, amphibians, animals and reptiles have salts, minerals and water in their body. People, birds, animals and reptiles lose water, minerals and salt when we sweat or go to the toilet (tinkles). Fish and frogs don't sweat but do lose some minerals when they go to the toilet, but to a much lesser extent than land animals.
Fish have a certain amount of water and minerals in their bodies and they have water constantly flowing through their skin, which can wash out the minerals in their body.
If you have fish from hard water or salt water and put them in pure fresh water, the pure fresh water will wash out the mineral salts from inside the fish and the fish can die from too much water in its body and not enough minerals/ salts.
If you have fish from soft water and put them in hard or salty water, the fish will take in a heap of minerals and salts and lose water. Then they become dehydrated and suffer from kidney problems and other issues associated with an imbalance in the minerals, salts and water in their body.
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Osmotic Shock in aquariums is uncommon and not likely to be the cause of your fish dying. It occurs within a few hours and the fish usually die within 24 hours. There's not really any symptoms for osmotic shock other than the fish acting unusual, not eating and being lethargic. However, these symptoms can be associated with poor water quality, stress from being shipped/ transported, etc.
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Unless your Bettas came from very hard water, there should be no reason why it would have problems with osmosis when put in reverse osmosis (R/O) water.
R/O water has little to no mineral content and can have problems with the pH dropping suddenly. Bacteria also grow faster in soft water and chemicals are more toxic to fish kept in soft water.
Bettas naturally occur in soft water and should be fine in R/O water.
About the only reason I can think of that would cause the fish to die from R/O water is chemicals, which are more likely to poison fish in soft water vs hard water.
Have you tested the R/O water for ammonia, nitrite, nitrate, GH, KH, pH and chlorine/ chloramine?
Maybe the shop gave you bad water or the water containers are contaminated with something that is causing problems. Maybe their R/O unit is full and needs new cartridges.
