It’s closer, but not quite there yet. Your problem right now is that self.input_data_paths is never initialized when you call raw_file_names(). Also, the property name should be the same as what you call in the .__init__().
class MyOwnDataset(Dataset):
    def __init__(self, root, input_data_paths, 
                 transform=None, pre_transform=None):
        super(MyOwnDataset, self).__init__(root, transform, pre_transform)
        # same name as property
        self.input_data_paths = input_data_paths

    def _download(self):
        pass
    
    @property
    def raw_file_names(self):
        # hidden variable
        return self.__input_data_paths

    @raw_file_names.setter
    def raw_file_names(self, value):
        # affect value to hidden variable, this is called in the __init__ method
        self.__input_data_paths = value

Let me know if this works!