So one thing that would be cool to have implemented is being able to switch the main speaker mid conversation (maybe with a couple voice lines to make the switch a bit more immersive... or some other method to make it seemless). That would be a simple (probably) solution to many of the frustrations.
But it would also be cool if your party was considered as one entity and all the skills/traits/races would factor into the dialogue. For example, I've been changing shadowheart into a drow to get past the goblins in the blighted village but if you walk in through the front with another character, her drow form doesn't get noticed. If the system did a check and found that there was a drow in the party, I think it would make the dialogues more dynamic and make your party choice be just that bit more meaningful.
And I also agree with the above quote that suggests having extra dialogue options for the main speaker based on who is in your party (and their backgrounds, races, classes, etc.). But on top of that, there are some instances where a skill check is needed where the main speaker should be able to delegate a task, for eg. picking a lock, or asking someone else for a religion check. There are times when it may not make sense, for eg. when reading the necromancy book and having to make saves directed at the main speaker/reader/receiver. However, for the most part, I think being able to choose someone else to make a skill check would again make things more dynamic.