I am brand new here, so I wanted to make sure I don't step on anyone's toes by doing something that is frowned upon. As I understand it, a story that is marked as being public can be added to by anyone, beit one chapter or an entire story chain of chapters. Is it expected to somehow inform the original creator of this? Or let them know beforehand? Or can you just pick up the baton and write away?
While I'm sure some authors may appreciate the gesture, they have to choose for the story to be public when creating it, and thus probably want for more authors to add on to it. They have to make the story free game for anyone intentionally. If you feel unsure then there's nothing wrong with throwing them a pm beforehand, but I don't think there is a specific etiquette you're supposed to follow in that regard. If they don't respond, I wouldn't let that stop you from adding to it either; there's a lot of inactive authors on this site. Some authors even make it very obvious they want their stories to be added onto by making a bunch of dead end routes that only go a chapter or two deep. If it's a story like that you're wanting to add onto, then just go for it. Either way, we all want more to read on Chyoa. Please don't hesitate to publish something you really want to write.
In my eyes, "public" means you are invited and encouraged to write on it, so I'd agree to go for it. Often, the reason for someone making a story public will be to encourage writing on it from many authors with as few obstacles as possible, so your writing will likely be welcomed. One caveat; besides any obvious writing rules chapter the story author may have set up, you may see a set of additional writing guidelines appear for the story when you hit to add a chapter (and nothing will be posted until you hit to "publish chapter," so I encourage you to check). In that space, the author may have written guidelines or requests that they intend for contributors to adhere to. Whether they end up enforced in any way or not, those may offer extra insight. Or, you might not see anything, as I don't believe it's a mandatory field. No harm in checking before you get too far in your brainstorming or writing.
Generally speaking, a public story is encouraging other authors to contribute, normally authors add some sort of guidelines to the story, so check for those. Beyond that, there are only two rules that aren't obligatory but fall under "etiquette" in my oppinion. Don't add a chapter to the latest chapter of a branch unless that branch is clearly abandoned by the original author. You're basically sniping the story in this situation.(some authors don't have a problem with that though, so when in doubt just send them a message) If the chapter you want to add to was not written by the original author but another contributor ask that author for permission to add to it first.
I'd add: stick to the spirit of the story. If it's a dubcon genderbender story set in a fantasy world, don't suddenly have Lara Croft crash in her X-Wing to engage in a week long watersports reverse gangbang with the spirits of the entire Napoleon family. ...Or something. ...brb, writing new story.
If you want to add anything that would only appeal to specific kinks (i.e. foot fetishs, scat, NTR) I'd recommend having some sort of warning at the top of the chapter or in the chapter title, because I think pretty much everyone on the site would agree that being jumpscared by vore without warning is not fun. This also applies to things that aren't erotic, but still not something people will always want to see, such as, say, a reference to a national tragedy or a blatant racism.
Stories I’ve made public when I feel as though people really want to contribute and I can’t be reviewing each one or when I abandoned the story because I couldn’t figure out where to go next and other people might know better. There are also fetishes I’m not into that I know others are. And I don’t want people alienated (well except for some that I find very off putting). the only story of mine that still gets public submissions frequently is Costume Store where people are transformed and or mind controlled into the costumes they wear.
Generally, if a story is public, you’re free to add to it without asking. That said, it’s always a good idea to leave a comment or message the original author out of courtesy, especially if you’re making major additions. Some writers love the collaboration, while others prefer their stories to stay as they envisioned. Just be respectful, and you should be good to go.