Record audio from your microphone, watch the live level meter to make sure you're being picked up, then play it back and download it. Everything happens in your own browser; nothing is uploaded.
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Your browser will ask for microphone permission the first time. If the level meter doesn't move while you speak, the wrong microphone may be selected in your browser or system sound settings. Without the save-to-disk option the whole recording is held in your computer's memory until you download it, so a refresh or closed tab loses it.