Kids in the Bathroom...

Depends on if they were going potty or in the bathtub. I never left them alone until they could talk. Once they could carry on a conversation with me I would leave them alone briefly like to get pj's or something. If they were alone I would hollar to them and ask a question so they would have to answer so I knew they were ok.

For doing what?

Public bathroom or in your home?

A child can drown, in only about 1.87" of water, face down.

You cannot always "hear" someone drowning. It is a silent incident.
They cannot even call out for someone for help, sometimes.

Depends on what they are doing:

Bathtub time, around age 3 we leave the door open and I clean the kitchen or near by space that is with-in hearing distance, checking on her ever 5 mins or so.

For the potty, around the age of 4, but still I am with-in hearing distance.

Do you mean the bathtub? Or just in the actual bathroom... like going pee by themselves? For the bathtub, we let our twins stay in the tub for a few minutes together with no immediate supervision when they were about 2.5. We were just in the next room or down the hall - so within hearing distance. Now at almost 4 they spend the majority of the bath time in there by themselves. I use the time to clean up the kitchen (within hearing distance). For the actual bathroom alone... I guess about the same age when they were potty training and had a reason to be in there by themselves.

to do what?
Public or home bathroom?
In our house if you are in the bathroom you are there by yourself unless you are learning to potty- after that I waited outside the door in case they needed help but I am not a fan of other people in the bathroom while you are there.

DD is now 3.5 and around 3 was when I started letting her be alone in the tub for a few minutes while I did other things upstairs like put laundry away. I can easily ask her if she's okay and hear her reply back.

She's still not potty-trained yet though so we are still working on that one. :P

Depends on what they are doing. I think SD was 9 before we started letting her go to the public restroom without someone lurking right by the exit. (DH was a single dad so if she was in there too long, he'd ask women to check on her.)

I don't yet leave DD alone in the bathroom and it's typically off limits (for various reasons). I'm not sure when I'll leave her alone. Friend's 3 yr old fell off the potty so I think even then I'll be close at hand. I leave DD to get PJs when she's in the tub, but her room faces the bathroom and I can see her. I only started doing that recently.

I don't remember the exact age, but I do remember when my daughter was old enough to sit in the tub with me in the next room with the doors open, I used to ask her to sing (which she loved doing - still sings in the shower at 11 years old). I figured as long as she was singing, she was fine and I was always close by - as in on the same level of the house.

Leaving my 3 year old in the tub alone creeps me out, so I sit and read while she plays. I do leave her alone while going potty, (she's been kicking me out since she was 2 years 8 months.. ;p) but I am always keeping an ear out for mischief. If she's in there for too long I check.

I let my 5yr old take a shower by himself like a big boy. But if he and his little brother 3yrs are in a tub (or just the 3yr old) then I don't leave them for more than a minute if I can help it. But, with my kids I'm really preventing a splash fight or them getting out and taking my good shampoo or lotion to make bubbles in the jetted tub. We have had more than one foam party by accident. Funny but what a pain to clean up.