This is not an exercise in linguistics; this is a consciousness raising exercise (something I've always disliked, so I apologise for that). Take the example of the phenomenon sometimes called 'the stranger of the train effect'. It has probably happened to most of us. We get on a train and we sit next to someone, and by the end of the journey the person has unburdened upon us a great deal of personal information about their divorce, say, or their mental illness or their worries about their drug-taking children. Stuff that they might only otherwise tell their closest friends. Or possibly not even their closest friends: friends can be dangerous things when interests collide (I'll come back to this).
Are they therefore showing that they trust us? Not at all, because trust is a transitive verb. They probably don't care who we tell about their problems, because we can do no damage with the information. If they tell their friends, their friends could break confidence and tell other people, severely damaging the storyteller reputation in the eyes of other friends, colleagues and family members. By telling us, they feel better about themselves, for 'getting it off their chest' and may even recieve some useful advice (and it is cheaper than therapy). In order to trust, there needs to be risk. If I climb a rockface I might trust the person holding the rope to not let me fall, if I tell a friend my embarrassing problem I trust him or her not to put this on Facebook. In each of these cases trust is important because there are potentially damaging consequences if confidence is not kept. In the case of the stranger on the train there are no consequences, so no trust is needed.
Returning to online. People often disclose a lot more online than they might do face to face (this is sometimes called hyperpersonal communication). People usually only disclose with close friends, so does this mean that people become closer online? That they trust them more? Possibly not, because again the disclosures may not matter. They don't know you or your friends, or your work colleagues or your family, and if you use inscruitible user names then they are not likely to either. People don't need to trust to disclose, if there is no risk there need be no trust.