There are many definitions of both sympathy and empathy. And presumably people have constructed careers out of the debates. I think sympathy can be shown to people but empathy cannot - it can be felt, and your response to what you feel manifest in compassion for instance.
Some people are more empathic than others, that is they are more sensitive to their own emotional changes when they encounter someone in distress for instance. You may feel sad when a friend is sad, and so there is alignment in this. However, you may feel anxious when they are sad but it is still an empathic response. So yes, you don’t feel their pain but rather your own, if you are open to it. It applies to joy as well and that is a key point.
Sympathy is almost invariably with respect to negative states and loss, but empathy applies to any emotional state.