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.
Parent companies can be either hands-on or hands-off with subsidiaries, depending on the amount of managerial control given to subsidiary managers. BREAKING DOWN 'Parent Company' A parent company is a larger corporation that has significant ownership over a subsidiary or group of subsidiaries. These partially or wholly-owned smaller companies are controlled by the parent, to varying degrees; however, all parent companies, for the most part, own more than 50% of a subsidiary's voting stock.