The loan agreement lists the terms under which a loan is given. The bank is the lender, and any borrowing entity--individual, partnership, company--is the borrower. The loan agreement covers many of the same terms and conditions as a promissory note, although it is longer and more complicated.
The subsidiaries remain private. They are private subsidiaries of a public company. Think about it: if a subsidiary went public, that would mean that its shares were being offered to the public, instead of all being owned by the parent company, and so it would no longer be a subsidiary.
If a parent company owns a foreign subsidiary, the company under which the subsidiary is incorporated must follow the laws of the country where the subsidiary operates, and the parent company still carries the foreign subsidiary's financials on its books (consolidated financial statements). company whose voting stock is more than 50% controlled by another company, usually referred to as the parent company or holding company.