Does it store anything about my visitors?

Not from counting. Counting keeps one row per day with three numbers in it: views, people and sign-ups. If you ask new subscribers to confirm their address, the date and time each one confirmed is kept with their sign-up, and nothing else about them. There is no address, no user agent and no third-party script, so there is nothing to identify anybody with.

A repeat visit is told apart from a new one by a cookie, trcs_seen, that the browser keeps for the day. Nothing is read from it but the date, and the privacy policy paragraph names it.