Let me say that again: Twitter is not private chat.
Why do I say this? Well, because it seems there are people out there who confuse Twitter with services like Glassboard, and think that people they don’t know shouldn’t respond to their tweets. Or maybe it’s just people who disagree with them; it’s unclear.
There are a few important facts that such people need to be made aware of:
People who follow them may retweet their tweets. As a result they may very well be seen by people who do not follow them, who they do not know and who might disagree with whatever opinion they’ve expressed.
By default, your tweets are public. That being the case, tweeting is like standing on a soap box at Hyde Park Corner, talking loudly to all who will listen. You don’t get to pick your audience.
If you say something on Twitter (or indeed from a soap box at Hyde Park Corner), and someone who sees your tweet (or is listening to you) finds it interesting or controversial, they have every right to reply. Your “conversation” is not private in any way, shape or form; indeed, it is not actually a conversation.
If you don’t like the above facts, Twitter has a mode for you; set your account to “protected” tweet mode. At that point, you do get to screen your followers, who can’t retweet you.
Yes, there are downsides to protected tweet mode. If you don’t like the way Twitter works, and you don’t want to protect your tweets, post to a blog instead and turn comments off. Or use a private group chat system like Glassboard. Alternatively, you will simply have to live with it.
Finally, if you ask on Twitter why people are replying to you when you don’t want them to, and someone points out all of the above, there is absolutely no excuse for threatening or abusing them.