Do you have pictures of the area? Is it a thick layer of paint or was the original spill/drip cleaned up and you are dealing with the paint that absorbed into the concrete?
Concrete is porous, soaks stuff up. It's old, which means it's going to be tough. Pressure washing is your best bet, but you have to be extremely careful as you can cut the top, smoothed layer of concrete right off and expose the aggregate underneath. If it's a wide area, maybe you're better off coating the area instead of damaging the concrete. Nice outdoor concrete coating, sand or earth colors. Might dress the place up and solve the spots at the same time.