I own half a house with a family member, I hear a rumor that they are thinking of getting a mortgage on there half. Can they do this without my knowledge or permission?
If the property is registered in joint names then the bank will not be interested in lending money on the security of one owner's interest in the property only.
Perhaps the OP is worried that in the case of the other party defaulting on their mortgage, half the house will be reposessed and they would be left with no downstairs.