I remember before I actually went to Japan, I'd always kind of known that Japanese families bathed together. And given my cultural background (and ignorance) I'd figured that was like what I did when I was really little, having baths with my mom. This is not how it works.
Let me start by saying there's a really terrific reason for taking baths in Japan, as opposed to showers. They don't tend to have central heating in their homes. This means that when I was living with my first host family, I had days when I woke up in the morning with my nose hurting from being so cold; once I checked with a thermometer and discovered it was 3 degrees C in my bedroom. Unfortunately, this was also the period when I didn't feel comfortable with the Japanese way of bathing, and I froze myself silly by trying to take showers. After I moved in with a family that took time to explain things and be friendly, I was sane and used the bath.
Every house or apartment I've visited in Japan, no matter how small, has had a bathroom. The traditional Japanese bathroom is all tiled (or at least water-friendly) and has water sources on the walls outside the bath itself. The bather goes in, washes head to toe in the bathroom, rinses off, and then at the very end steps into the bathwater, which is mostly for relaxing and warming up. It makes more sense (at least to me) for people to share the same bathwater when they're each getting clean first.
Some tubs have ways of keeping the water warm, and some don't. All the tubs are deep, though, because they're made for soaking, often up to the neck. I think the funniest thing that ever happened to us when my husband and I lived in Japan was that we each used the old-style bath belonging to some friends who lived on a tiny island off the north coast of Honshu. I was shocked to discover that the longer I sat, the hotter it seemed to get. I had no idea what to make of this, so I took a very quick bath; then my husband discovered the same thing. When we asked what was going on we were told that there was a real fire burning underneath the bathtub! Now I know what a lobster feels like... To make this particular bath pleasant you had to keep pouring in loads of cold water. Little did we know.
Then there are the public baths. These are mostly divided into men's and women's, but are not always. (In some natural hot springs you can even bathe with monkeys - now there's inclusiveness.) Many people will take a washcloth into the tub with them, to cover up critical areas, but in general I haven't noticed a great deal of embarrassment among the people who bathe together in this context. People can in fact be very friendly. I was adopted once by a group of elderly ladies who decided that as a foreigner I must not know how Japanese baths were supposed to work, and took it upon themselves to teach me. Because they were sweet and solicitous, I let them teach me even though I'd been through the directions a few times by then.
There is in fact some degree of modesty in the public bath context, but the taboo isn't about being naked. It's about having someone else see you get undressed. Outside the common bath area there are generally curtained stalls for the undressing part.
I decided to talk about this today partly because I found it culturally interesting, but also because I think it shows the degree of cultural difference that is possible surrounding a single activity. You see architectural differences in the bathroom; different shaped tubs; different ways of heating water; a separation of the function of the bathroom (washing) and the bath (soaking); different rules of behavior including gender separation, context for modesty, etc. This doesn't even include the rules about who gets to go first/next/last in the family bathtub.
So if you're working with an alien or fantasy culture, try putting some thought into the various details of activities in your newly created context. Including a "bath scene," or a scene that shows another common daily activity in detail, gives you a terrific opportunity to deepen the culture you're sharing with the reader.