I've been in lots of bands and it is indeed always different. The band I'm currently in consists of 3 Classically trained musicians and a relatively self taught singer/songwriter. While we have the capability to write everything down, we primarily don't. The one song that does have some parts written out was when our guitar player wrote the song instead of the singer/songwriter and that was basically because it was very riff heavy and made it much easier to learn outside of our practice time. So we rely on memory for all of our songs. I think this is good in a lot of ways, particularly not having to rely on a piece of paper with notes on it to know what to do. I also think that it allows the song to sink into your subconscious a little better and allows you to focus more on playing expressively. However, this does become a more difficult task when you expand your band's catalog. We've been playing for years and have a lot of songs that we don't play much anymore and when we decide to pull them out of the closet, it is sometimes incredibly difficult to remember how the song goes, let alone what exactly I used to play. But that's where extensive practice comes into play. We used to practice those songs and perform them very, very regularly, so half the time I can't think of what it is I'm supposed to do and then I'm suddenly doing it.
Other rock bands seem to conform to this approach, though there are surely exceptions. I tend to see more things written out when it's a Jazz band or if there is a horn section or string section. A lot of the rock tradition comes from people without a formal education, so they typically either don't know how to read music notation, or prefer not to. Every other band I've played in consisted mostly of players that couldn't read music, so we wouldn't have been able to thoroughly notate everything if we wanted to (unless I did all the work and played everyone's parts back to them when they needed a refresher).
So I think the answer is not only going to be specific to individual bands but also the individuals within those bands and often times the genre of the band. The important thing to do when you start playing with others is to come up with a way to make sure everyone is on the same page with their varied experiences. Recording a simple reference track is probably the easiest way to document things but if everyone can read, it's usually easier to have things written down (maybe not note for note but key sections or specific ideas) than to pull up a recording of a song and find the section you're looking for, especially if the songs evolve over time and you would need to record new reference versions repeatedly.