So, one observer will see those oscillations happen faster than the other?

Not quite. In each observer's frame of reference, time appears to pass the same; it's only when you try to reconcile the between two objects that are not at rest with respect to each other does relativity show up.

Basically, when you bring someone back to Earth, the observers will find that their watches don't match up even though both observers experience time passing the same way as normal (because the oberserver is by definition at rest with respect to their own frame of reference).

TL; DR: Relativity is a pain in the ass and makes no sense in everyday terms.

edit: disclaimer - I am not a physicist and have not taken physics classes in a decade plus, but I do teach science at a college. I'm going mostly on half-remembered lectures and some random one-off discussions I've had with my buddy in the physics department over the past few years.