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.
Many of the largest private companies today have been owned by the same families for multiple generations. Staying private means the company does not have to answer to its shareholders or choose different members for the board of directors. Some family-owned companies have gone public, and many maintain family ownership and control through a dual-class share structure, meaning family-owned shares have more voting rights. Going public is a final step for private