reff : Another Earth Explained

Yes, it's most likely that Rhoda of Earth 2 has come because of John from Earth 1. This is clearly not a case of John from Earth 1 and Rhoda from Earth 2 crisscrossing and landing up in their "other" Earth in the same time frame.

Look at this picture:

That's Earth-2 as seen from Earth-1.

How far does that put the two Earths? Not very far. In and around the distance of the moon. It takes 3-4 days to get to the moon from Earth - this is for a manned mission. Let's triple the time, that would make 12 days, about 2 weeks. 2 weeks is how long it would take to reach the other Earth.

Rhoda of Earth 1 and Rhoda of Earth 2 run into each other after months and not weeks. That removes the possibility that Rhoda of Earth 2 and John from Earth 1 leave their respective planets at the same time. Rhoda of Earth 2 arrives much after John of Earth 1 reaches Earth 2.

Perhaps there was no competition on Earth-2. Or even if there was Rhoda didn't win it as she arrives only 3-4 months later.

Hence it is more likely that John's landing on Earth-2 lead to Rhoda of Earth-2 getting to know the story of the accident and influenced her to come to Earth 1. If not for John's story, Rhoda on Earth 2 would be one among thousands of MIT (assuming the accident doesn't happen) and other top university students enthusiastic to travel to the other Earth. Not enough to be picked for travel.

If the accident did happen on Earth 2 as well, Rhoda has no reason to be selected if not for winning the competition (which she hasn't else she would arrive on Earth 1 much sooner, in 2 weeks and not 4 months)

Hope that clarifies.