"Empathy" is not just a synonym for "sympathy." A true empath is one who experiences the emotions of others as if they were their own. A true empath senses an emotionscape all around them, often "seeing" individual emotions as rays or waves.
Physical distance does not matter. The strength of an emotional signal depends on three main things: how strong it is at the source, whether the source is thinking about the empath, and how well source and empath know each other.
It can be difficult to tell the difference between an emotion from within and an emotion from someone else, especially without context - they fundamentally feel the same. Also, the empath's mind will try to supply some kind of reason for any unattached emotions, which can be further misleading. But each individual person has a signature feel to them, borne of their souls, and each of these is unique. It is possible to discern these signatures, given enough practice.
In real life, emotion waves are just one of many aspects of a given social interaction - along with body language, tone of voice, and words. Online lacks much of the nonverbal context of offline, which actually makes the emotional signals easier to see.
Empathy is not the same as telepathy - the ability to read thoughts. Knowing what someone feels is separate from knowing why they feel it. However, given enough context surrounding an emotion, an empath can often deduce the thoughts with amazing accuracy.