I do find it a bit annoying. People will chase me down to talk at length about a problem, wanting me to give them advice I'm not at all equipped to give and that they won't follow anyway, and just by their speech and their example ('I was awake all night thinking about it, I can't stop thinking about it') a non-empath could figure they were anxious about said situation. But I can feel it vibrating unpleasantly in my chest, and finally after standing there, somewhat trapped, kindly listening, etc, I might say, 'You're very anxious.' And they'll say, crossly, indignantly, 'No! I'm not.' Argh. Then I go home vibrating.
Mostly I try to maintain good boundaries and not mention people's emotional states etc, and just note them to myself. I find writing about it afterwards clarifying and somewhat soothing. I can validate myself nowadays. The days of me wanting to be the world's Agony Aunt are also over.I think, in answer to your question, my take on it is it not really your/our business - their emotions - except in so far as it affects you/one, personally. Most people, as others have said here, are not really aware of their emotions, or are in denial about their emotions, or are out of touch with their emotions and their bodies. That is also not your or my job to address, in my view. (They'll probably notice they feel better after talking to you, and you may set up a vicious cycle where they'll ignore you, dump on you, feel better, and leave you like a worn out rag, or overfilled sink. Then it becomes,"Physician, heal theyself!")