Your question is a bit tricky. If say So. Cal. is around 35 degrees North of the equator then a person opposite of you on the earth would be at 35 degrees South. They would not be observing the same stars 12 hours later from their vantage point. Now if opposite of you means the person at 35 degrees North, then they would see the same stars 12 hours after you do.
Yes, I just referring to opposite on a rotational line. What I was really asking about is,
as we circle around the sun for 365 days, how much of the milky way do we see.