I'm a sensitive person. Thanks to a rough childhood, I tend to be hyper-aware of people and their emotional states. As you can imagine, this sometimes stresses me out. There are a few studies that hint at the interplay here:
Anxiety can reduce empathy (speaks to an anxious state)
Anxious people tend to be more empathetic (speaks to the anxious trait/personality)
Here's a great explanation of these seemingly contradictory studies by the good folks over at the Greater Good center.
"They found that stress-prone people were good at cognitive empathy—in other words, accurately identifying inner states based on outer clues. But there’s a critical caveat, for the purposes of our discussion: They weren’t as good at 'affective empathy.' That’s a science-y way of saying that they could recognize an emotion, but they weren’t necessarily feeling it themselves.
This makes perfect sense, in the context of the research to date. Stress mobilizes the body’s resources to survive an immediate threat. Among other effects, it helps narrow our attention and zero in on the threat. If you’re prone to be socially anxious, meeting strangers stresses you out.
That’s why anxious people can appear to be shy; they’re simply avoiding stressful stimuli, often going deep rather than wide in their social networks. Walking into a party or asking for help from people can take enormous courage. In those moments, their bodies are flooded by hormones that help them focus on threats—threats that are embodied in the faces of other people. This helps with cognitive empathy.
But I bet the reason why their affective empathy goes down is that they’re momentarily denying themselves access to their own inner states. Their attention sharpens and goes outward, which makes perception more accurate. But at the same time, they’re instinctively protecting themselves from getting caught up in the feelings they detect. This might help make socializing emotionally manageable. It might also make them seem cold or just a little stiff, in addition to shy."