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No, the person you are talking to cannot understand who you are unless you reveal your identity during the conversation.
You can block the person you are chatting with from the chat menu on Derdost and ensure that they never match with you again.
Each conversation on Derdost lasts 24 hours. However, if both parties pin the chat, the conversation time limit is eliminated.
You can only be in 25 unpinned chats at a time on Derdost.