I doubt that. There will be those people that just never stay on one planet. Travelling around the universe constantly.
Basically keeping the diseases relatively uniform everywhere.
There might be some diseases mostly eradicated in one place that keep cropping up in another, just like on Earth, but for the most part constant contact would probably keep each planet up to date with the other's vaccines. If the two planets lose contact for an extended time for some reason, then yeah, it's possible for people to become less immune to the other planet's diseases. It would be a really, really, really long time before the populations evolve into separate species though, so once contact was re-established and vaccines started flowing again it would probably even out.
Evolution is an amazing thing. Like bringing a mongoose to an island filled with snakes, the mongoose is an invasive species, has no natural predators in that area, and wipes everything else out. The disease is the mongoose.
It's very possible, there are tribes in the amazon where it's a real risk for them to be exposed to any other people because of their lack of immunity to diseases that are fine for us. They've been isolated for too long.
Basically keeping the diseases relatively uniform everywhere.