Hi,
I'm tremendously interfering.  I feel ashamed about it too.  Anyway, you are entitled to remove messages from your talk page (and only your talk page - do it anywhere else, and unless the text you remove is SHITSHITSHITCOCKSHITCOCKSHITTYCOCK, you'll get a warning and disapproval), but best practices is to archive.  There's even a handy  that'll do it for you.  By removing parts of discussions in particular, you run the risk of interrupting the flow of a discussion.  For you and your conversation partner, that's not a big deal.  For anyone else reading your page, it can be (particularly if you're discussing an article - often user talk pages become impromptu talk pages if two people are debating an issue).  Yet more of wikipedia's rules and mores that are not immediately apparent to new users.   (talk) Wikipedia's rules:simple/complex