The early definition of privacy was "the right to be let alone". But since the late nineties of the nineteenth century, a lot has changed. As we create a bigger and bigger "data shadow" just by living normally, it becomes more about "the right to claim ownership of data about oneself". And data is a very malleable concept these days.
In short, we should take the means to try and keep the data about the site visitors to the minimum, to ensure a working site. We need not even store personal data; in most countries it is quite possible to match a person to an IP-address. Moreover, we should not gather more information about our visitors than necessary to process their requests.
But why, I hear some of you asking, shouldn't we gather the information? There is nobody other than us to read it anyway. In general, this may be true, but where I live, numerous illegal lists with personal data have made it into the news, while companies which got broken into because of shoddy security had to...