Sometimes it happens that I disagree with someone or maybe there is little understanding because of the complexity of the matter. What should I do? What is the proper way to handle discussions?
While they are very useful, comments are third-class citizens in the StackExchange network. They are meant to be used for short and temporary things. They must be used wisely.
When you're commenting and you realize the comment thread is getting too long, move the discussion to chat, either by using the option that appears, or by manually inviting the user to chat; whether you open a new room or use the main one, is not important. :)
Don't use comments for long discussions or better, don't use them for discussions at all.
By the time you realize that you have posted so many comments to and fro that they might obscure other comments for casual readers, it is too late anyway, so I don't think it really matter a great deal any more by then whether you stop or continue. Other than the (minor) point that they can be a bit cluttersome for casual visitors, I don't really think discussions in comments are bothersome in any way.
On the other hand, they can be very interesting, constructive, and satisfying. Some of the best parts of certain questions are in comment threads. So I don't think there is a real problem here: I am perfectly fine with discussions in comments. It would be nice if they were actually threaded, though; that way, casual visitors could easily skip long threads, and the problem outlined above would disappear. I wish SE implemented this.
As to moving a discussion to chat, that usually doesn't work. First of all, many invited users just won't go to chat. They don't want to or don't feel the need to or whatever. Secondly, other users won't see the chat messages in the comment thread, so they will not notice, and they are even less inclined to go to chat, so if any discussion happens there at all, it will nearly always be between two people.