The early Buddhist texts (the canonical suttas) describe meditations and trainings to develop this skill. The reason the suttas devote so much time to explaining these techniques is because they are NOT common knowledge.
The etymology of compassion means "suffering with". It's baked into the word (in English) the assumption that compassion involves suffering for both parties.
In Buddhism, we learn that compassion doesn't have to create suffering for the holder of compassion. We learn that it's possible to dwell in the state of compassion as a way of moving out of suffering, rather than moving into it.
I am grateful to you for pointing out this etymology. I find it to be beautiful because in compassion I can be suffering with Christ. So, thank you.