From what I have observed, I might be wrong, it seems that people don't necessarily.
If an expression of empathy is attempted accusations of 'minimising' can ensue. Because it is not the same, exactly the same but what ever is? If you empathise emotionally, then again, not wanted because you are trying to shift the attention onto yourself.
Even worse if you have overcome the particular problem and talk about how.
There must be a balance in this because people will accuse others of not having any empathy. But what do they want? A hug - no, not necessarily, some people hate that. Join in complaining? No, makes them feel worse. To listen? (Do you really need to be there and not respond at all?,) - well it doesn't help.
So, I do feel empathy. But should I express it? How?