I'm pretty sure they are all privately owned by the landowner (usually a sporting estate, although some are in areas owned by conservation groups e.g. Ryvoan) - even if they are maintained by the Mountain Bothies Association.

NB The default in Scots law is for any building to belong to the landowner even if someone else builds it so even if someone did build a hut somewhere it would still end up being owned by whoever owns the land.