If I'm with a person who unintentionally did something embarrassing, I feel embarrassed for them. If it's a stranger, I just get anxious about how embarrassed they might be feeling and so I try not to stare or do anything to embarrass them further.
However, if somebody is intentionally embarrassing themselves but don't care, I don't get embarrassed. Most people do if they are with somebody who is showing themselves up. But I just think that it's them who's drawing attention to themselves, not me. If I just act calm and casual, then they know that I'm not the one embarrassing myself. If I start to swear or snap at them, then I'd be showing myself up too, because it'd most probably turn into an argument. So I don't argue or swear or rant, etc. I just stay cool and calm.
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Female