Is Apple Keyring data stored locally-only on devices? I would like to understand if the keyring database is stored only on the local Apple device or if it is also stored on servers on the internet.
As Mike Scott wrote, depending on your iCloud backup.

See "Set up iCloud Keychain".

See "Keychain data protection" on page 20 of the "iOS Security Guide" for how the keychain works, how it protects access to keychain items so they are only available when the phone is unlocked, using the secure enclave processor.