From what I've been able to tell, if you've moved, acted or even selected a shared initiate character *at all*, then clicking the end turn button on any other character in the shared initiative makes the game assume you're done with all of them. It will switch to a shared initiate character when you click the end turn button only if you have not interacted with that character at all - not even clicked on it. It's poorly implemented, even if it's a neat idea. The display gets out of whack easily as well, and will often show characters at different positions in the initiative when they are actually sharing, and sharing when they are not. Sometimes removing other intermediary enemies from the initiative order will cause characters to begin sharing ,and sometimes it won't. The mechanics of this, as well as the actual intention, are distinctly unclear at the moment.
For some reason, if you revive a downed character that you are sharing initiative with, they lose their action, while, if they are just 'after' you (no enemies in between, directly next), but not 'sharing', then they get their whole turn. This is because you lose your action at the beginning of your turn, if you start it downed - something that shouldn't happen by the rules, since if you nat20 your death save you can take your conscious turn immediately, in PnP. Ideally, if you are sharing initiative with an ally, and they revive you with their action, then you haven't ended your turn yet, and you shouldn't have lost any actions.
