Regarding such friends, I think you have 2 big options:

1 : Ask the other person directly if they wish to take it off the chat room. Many people are hesitant to do so because of security. Some people may simply love chatting through chat rooms and may prefer chat room to everything else, resulting in them not focusing on security alone or at all, but on their preference. You can maybe ask such people to chat with you via personal messengers like MSN Messengers or other places.
2 : Forget about it, like you said. In this case you can continue talking via the chat room. Like I mentioned above, security is very important in such cases in the minds of many people. You do not know completely who the other person is, nor does that person know who you really are.

Also, the kind of chatting you are doing can play a great role in finding out more about the other person. For example, if you are in a gardening or spiritual chatting service, e-mail exchanges [stereotypically] may be less obvious and common than in a gaming or dating chatting service. What do you think?

Yes, a lot of headaches are floating around the internet. I think there can be a thin line between people who bring offline frustrations online in order to get rid of the frustrations without burdening others, and people who bring offline frustrations online in order to dump it onto others.

What do you think?

Karah, thanks for the comment.

I am guessing you have experiences some or all of the points online?

Chamonix, thanks for the comment. That is a good point.

Many businesses and business people are starting to call their prospective customers “friends” in order to add some kind of a personal feeling to things. Do you think such a trend can result in more friendship, or less friendship?