Pickled eggs do go bad if not refrigerated and so you should keep the pickled eggs in the refrigerator if you do not eat them or serve them within 2 hours.
That is especially true for homemade pickled eggs that are not in a sealed container.
If you buy store bought pickled eggs that are sealed up in a jar then you can leave those pickled eggs out at room temperature until you open the pickled eggs.
If you then open the jar of pickled eggs you need to refrigerate them until you use them.
The pickled eggs should last for at least a month if kept refrigerated.
You can leave the pickled eggs out at room temperature for 2 hours and after 2 hours you need to refrigerate them.
Bacteria can grow on the pickled eggs when left out at room temperature for too long so make sure to keep the pickled eggs refrigerated.