No, the user can´t edit all fields. The two fields that I marked in red on the attached image are impossible to edit if you´re logged with Twitter or Facebook account first. The cursor does not appear on them and I cant write anything. I do not know if it's an option that I have verified or smth like that. If I do the opposite (if I register first, edit this two fields and then add the Facebook or Twitter accounts) no problem. The items still appears.
First I thought it was a problem of the browser (safari) but I tried with Firefox, Chrome, Opera and the problem is the same.
It is important to fix this problem because in these two fields depends the access to restricted areas of the web.
Thanks again.