Cache-Control private says that it's OK for the client to cache the page, subject to its expiration date. The expiration can either be provided with Cache-Control: max-age, or with an Expires HTTP header. In the default case, the page is set to expire immediately, which means that it won't be cached.
One of the purposes of Cache-Control: private is really to tell intermediate proxies that they should not cache the page.
BTW, just because a page is dynamic doesn't mean that it should never be cached. There are many cases where caching a dynamic page is appropriate. You can cache not only at the client, but also in proxies and in the server's output cache.
(FWIW, I cover this subject in detail in my book).