I don't know what someone with a paid account can see. But someone with a free account -- which is most people, I think -- would not see your IP address or a list of the other profiles you looked at, even if you were looking while logged in. If you were logged in, they would see that you had looked. If you were not, I believe they would see "anonymous user" or something similar.
All that said, your nervousness makes you sound weird. Why would you care that a colleague or someone you planned to do business with knew you had viewed their profile? If I were a professional dog walker, say, I would expect (and welcome) potential clients viewing my profile. Sometimes I see that former colleagues have looked at my profile and I think that's interesting (as in, "Maybe they are looking for a new job") but I'm not bothered.
If you are ashamed of them knowing you look, then don't look. It's not like there is juicy stuff in a LinkedIn profile, anyway.