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Yes, it is permissible to speak during the athaan, or after it, no problem.  However, the Sunnah is to pay attention to the athaan quietly and respond to it.  However, there is no harm if someone needs to speak during that, while he responds to the athaan.  There is no harm if he returns someone's greeting of salaam, says "yar-hamuk Allaah" to someone who sneezes, or requests something he needs.