WordPress REST API calls that depend on the WordPress User

You could hook into the login process in the WP site and create a unique token (stored in meta) for that user on login. The token can then be sent to the other server, that could query the WP API with that token.

You would have to create a custom endpoint that validates the user token. Probably best to have some hash for the userid also.

Furthermore, on logout from the WP site or in some token validation cronjob, this token has to be invalidated for security purpose. Even better, if it’s just one request from the ‘other’ server, invalidate immediately and if on the WP server again, create a new token. So in effect, it’s a CSRF token (nonce) over multiple servers.