I just had my baby on Saturday night. There was VERY little mess. Hardly enough to even pay attention to, actually - and I did birth on the bed (hands and knees).
We have 2 diaper pails - one in the bathroom, and one next to the bed. Everything goes in the diaper pail - fluid-soaked diapers/pads (my water broke about 12 hours before labor started), cloth wipes, undies with blood on them, any towels we used that got fluid or blood on them, baby diapers, bloody diapers (what I used for postpartum pads the first day), all of that kind of stuff. And then I washed it all together, just like I do my diapers - one wash cycle on cold with detergent, then one wash on hot with a half-measure of detergent, baking soda, and about 15 drops of lavender oil. Everything came out clean.
I did use chux pads on the bed - I sat on them, kneeled over them, and what have you. If I had known the baby was coming so quickly, I would have put a couple fresh ones down right before pushing....but as it was, he came so fast that he was birthed over a single pretty-soaked chux, and a small puddle of amniotic fluid spashed onto the bedspread - maybe about the size of a dinner plate. (It's still there - I really could care less about a slightly blood-tinged fluid patch on my bed.....but that's just me
I asked for a fresh bedpad right after he was born, wrapped him up in a couple recieving blankets (which got changed as they got damp, and those went in the diaper pail, too.), and then when I felt the next contraction coming, I asked for a basin to squat over, and birthed the placenta right into that. No fuss, no muss
We kept the placenta in the basin nearby, and cut the cord a few hours later, before going to bed. DH is *supposed* to wrap the placenta in a plastic bag & stash it in the freezer until spring, when we will bury it outside. But we don't have any plastic bags right now, so the whole basin is sitting in the freezer :LOL
I think changing a poopy baby in the middle of the night is tougher to handle than "birth mess"!