I think you are misleaded by the branding on your debit card. Most transactions are not going through Visa, but a third-party solution that the banks them selves run. I might be wrong, but I think Nets delivers the solution in most of northern Europe. They call it NetAxept.
This is a direct pay solution used with online terminals, the Visa/MasterCard-part of your debit card is just for offline transactions or credit.