I live in a two story condo on the bottom floor. Today, I noticed my bathroom sink was filled for no reason. Every time I would scoop out the water, it kept refilling. After some troubleshooting, we realized that the backup occurs when my upstairs neighbors turn on their bathroom sink which is right above me.

What needs to be done to fix, and who is responsible for calling the plumber? Is the clog in their unit or mine?

You have a blockage in the drain after your sink. Blockage could be anywhere between your sink and the main drain line. This would be responsibility of the condo association.

This happens all the time in multi-story condominium buildings. You share common drain for that sink with your upstairs neighbor. As Harold said in above post, the blockage is somewhere below your sink. You have to "snake" that drain

Milo