Some faucets have a gasket so plumber's putty is not used. It just has to be centered in the hole (assumng hole is correct size).

As for the faucet, you have to figure out the exact source of the leak before you can decide. Might be a bad fitting, or it wasn't assemble completely, or something needs to be tightened. This is often the case for pull out spray faucets. If you have that, pull it out and see if the hose is screwed on snugly. When they are loose, water goes down the "hole opening" and into the cabinet.

I see. I took a look at the connections of the arc to the pipe inside the cabinet. It does not have the white plastic nut to cover the hole, while the cold water, hot water, and the spray pipe do. I was trying to touch where the water is coming out. It seems to be the big nut which is kind of inside and flushed with the hole and is connected with a small nut below.

Only you would know the answer after the facuet is removed and inspected. You say it looks and works fine,why would anyone say different when they can't see it? Now if you are asking about paying a plumber for labor,why not ask him since he can see the facuet. I have removed many facuets for various reasons and reinstalled them without problem.