Periods generally end sentences, and if that sentence convenes a thought, they do conclude the thought. I consider that to be the default mode of conversation, regardless of whether it's chat or mail or whatever. I like it when people take the effort to bring their thoughts to a conclusion rather than flooding me with cheap stream-of-consciousness-style conversation (cheap for them, not for me), and I tend to treat others the same way I want to be treated.

If I want to explicitly leave a thought open for continuation, especially in chat style communication, I place three periods at the end of a sentence. I consider that an invitation to the reader's mind to conclude the sentence. So it's either one period or three of them for me, but none at all just feels wrong.