My personal feelings, that aside, is that though many of us are privacy conscious, adding more and more dependencies to your site results in us having to trust more entities. Even if they don't store anything, we have to trust they aren't lying, that redirection is implemented properly, etc.
I think the best thing you can do if you care about the privacy of your users is minimize the amount of information necessary. So if your site doesn't require email, don't take it. If a phone number isn't necessary, don't ask for it. Use usernames, only ask for an email when the user is doing something that would require it (e.g. they need a receipt).
One thing that I love is when a site actually gives you a temporary username the minute you visit the "app" portion and you can use the site as if you created an account without having to do anything. That's usually a sign that the administrators really do care about you not jumping through hoops.