Actually, you are both correct. An unquoted reply can be in response to the OP or the previous post. In the case of a string of exchanges between people, quoting each other would only occupy excess space and would be unnecessary for continuity. However, one can also reply directly to the OP without quotes because the general thread topic surrounds the OP topic. It would also appear funny to quote the OP because everyone has read at least the OP.
This is where reading comprehension, context clues, and critical thinking come into play. If an average reader possesses at least two of these skills, then he or she can typically determine the flow of the thread and make an educated deduction about the meaning or intent of the post in question, or at least infer from it what the poster intends to convey and to whom he or she is replying.
