No critters in my open compost piles - just worms.
I bury the kitchen scraps - and chop into smaller bits to help it compost faster.
Wetter is better - downright soggy.
Everything I've read indicates that a compost pile should be moist, but that any excess water should be able to drain away. Don't put your compost material in the lowest spot of the yard.
If rain is not providing your compost pile with enough moisture, you will need to add some water yourself. Collecting rainwater is a good practice, because rain is generally close to pH neutral and has no added flouride or other chemicals.
I do not add additional moisture to my heap unless there is an extended drought. Simple reason is that I don't have time to worry about it. I'll use it when it breaks down, but I'm not really helping the process along as much as many folks will.