should you make them look stupid as they obviously deserve?
The simplest answer:
"Never argue with a fool. Onlookers might not know the difference." - Mark Twain
More . . . I'm going to wax philosophic as usual. When I start feeling this way, I think of a few things . . .
Don't grab a sleeping tiger's tail. There may be things you're unaware of, or forgotten, or that can be revealed to turn the whole mess back on you.
You say "obviously deserves" but ask yourself, what, and who, determines someone deserves something? From what perspective? Did someone assign you (the rhetoric you, as in me, you, I anyone) as judge and jury to determine what is right, what is wrong, and empower you to sole out appropriate punishment? The answers are always no one and no. :-)
I'll use a couple examples: people who lie about their income, knowledge, skills, or training, or even just inflate these, are doing so because of some deeper insecurity. I know a lot of people like this. Drag them from under the rock and show their underside, one or two things is likely to happen, a full blown countermeasure or it can even be devastating on severe levels.
How much energy to you want to put into this, and truthfully, what is your motivation? (Rhetoric "you" again.) It boils down to your own need to be right. Remember the cartoon, "Come to bed," "I can't someone is WRONG on the Internet." Same thing.
The redeeming part: these kinds of problems will wind themselves down without help from anyone. Karma always pays out.