+1 to speaking with PTs and sports docs.
My additional two cents:
-Make sure they know what activities you want to get back to (e.g. lifting, sports, etc). When I had shoulder surgery a handful of years back, I asked the surgeon what the recovery time would be. He said, "12 weeks to full activity." What he
should have said, was "12 weeks to lifting a 5 lb dumbbell." I wanted to get back to gymnastic strength training. It was a good two years before I stopped having residual soreness and discomfort.
So I am a big advocate of pushing for information and clarity from your surgeons, doctors, and physical therapists. Make sure they know what your goals are, and make sure you find someone to help you get there.
Outside of that: there is no harm in training the non-injured arm. Studies have shown that training only one limb results in strength gains in the untrained limb (though slightly lower), so it may even help you come back to pre-surgery strength faster once you're healed up. You can do legs of course, and core work that doesn't involve the recovering limb. The biggest thing is to make sure you don't involve the recovering limb too much in any way that might hinder its recovery. That's a question for a physical therapist
Hope it helps, happy healing.