Does Beth leave or stay with the family?

I'm convinced it was left vague on purpose.

A recurring theme in the show is to hint at potential plot points in the future, while leaving the possibility open that it is just ignored. Dan Harmon has stated before that they don't really plan out the future details well in advance. They leave multiple different options open until the time comes to use a plot point. Then it becomes canon.

They want us to question if Beth is real or a clone going forward. That said, if Beth is a clone, I don't think we will find out for sure until there is a good reason to reveal that information. They now have another back-pocket dues ex machina where "real" Beth can always come out of nowhere and change any situation.

Rick also makes it pretty clear that the audience wont really be able to tell either. He says "Either way, you chill the f** out" (or something like that); meaning we can't definitively say Beth's new behaviour is from cloning or just a new outlook.

The ambiguity is essentially confirmed to be on purpose around 1:30 in this video from the writers/creators:

Also in Season 3 episode 10

Beth has to question this for herself, and it leads to the secondary plot in the episode.

Season 4 Update: It seems that as of season 4 episode 10, we can now give this as definitive of an answer as you can get.

Season 4 Episode 10 Spoilers:

The episode starts off with a very different looking Beth in space, doing very Rick-like things. At first it seems obvious that Beth cloned herself and left. As the episode progresses, this becomes less clear; since the Space-Beth appears to have a bomb in her neck which will explode if she comes back to Earth. The episode leads to both Space-Beth and Earth-Beth questioning if they are the clone or not. Rick gives very unhelpful answers all episode.

At the end of the episode, we find out that:

even Rick doesn't know which is real. Not only did he erase his own memory of what happened in a Morty's Mind Blowers way. When he goes back to the memory, it is revealed that Beth asked Rick to choose what she does. Rick then made a clone of her, and shuffled the choices in a way so even he didn't know which was which.

So as of Season 4 Episode 10, it is now established in canon that:

We literally cannot know if that Beth was a clone or not; but we know there is a clone of Beth. We just can't be sure which of the two it is.

At this point it is petty obvious, Beth is a clone. First off space Beth must have gone to explore the cosmos and Rick cloned her to have stay on Earth with the family. Obviously Beth didn't clone herself cause she would remember and the episode would be nonsense. Anyway, when the clone of Beth was given the same option on if she wold want to leave or stay with the family, giving her the idea that if she was a clone, Original Beth would've left to travel the cosmos. Clone Beth is skeptical and wants to find out the truth. Meeting Space (original or space) Beth confirms that (clone) Beth is a clone indeed, since Space Beth had returned from exploring the cosmos, although clone and space Beth don't catch n, which allows Rick to play with their minds and convince each other not to kill the other implying that the other is the clone and not themselves, which leaves them unhinged and alive. In short, Beth went to explore the cosmos, was cloned, the clone had the idea of possibly being a clone, Rick tells them not to kill each other using confusion, and space Beth and clone Beth continue on their separate ways.