I used one from a hire shop and to get the carpet clean I found I had to really wet the carpet (in fact i went over it with a watering can which might be a bit extreme, but the carpet was pretty manky from previous occupants.
Dirt won't come out on its own, needs to have the shampoo and water to carry it out if that makes sense. I hadn't time / patience to go over it several times (the basic concept of the wet vac is that you spray out water /shampoo mix from the vac, then suck it back up).
Anyway, this got lots of dirt out, the waste water was BLACK. Wasn't 100% by any means tho, carpet still stained. MAybe another run would have helped.
But the carpet was damp for a couple of days, left lots of heating on and alternatively left doors and windows open.
None of that is brilliant if you have a toddler.
Suggest you buy / hire a dehumidifer as well. No point turning up the heat if you are not getting the damp out of the air / house.