Just as the question stated. I know it is marked under the private section of our profile, but I was just wondering if you could see stuff like that. And if you don't, do you see anything else that a normal/regular user wouldn't see?
Yes, moderators can see your real name and email, although this information isn't something we see on your profile by default, we have to click several buttons to see it. We also have to agree to abide by the Stack Exchange Moderator Agreement when we become moderators. This states:
ii. I acknowledge that I may have access to potentially personally-identifying information about Ask Ubuntu users and that in connection with such access
b. I will not disclose this information to anyone,
c. I will not store or copy this information and
d. I will only use such information in connection with performance as a Ask Ubuntu moderator for the benefit of Ask Ubuntu.
So you can be sure that your information is pretty safe.
As to your question about other things we can see, moderators have access to all privileges as well as various other tools used in the moderation of the site. If you have any more specific questions ask away.