Are vectorized posterior distributions independent?

I was running some simulations in which I needed a separate parameter estimate for each run of 1000 simulations. So I was basically running a for loop in R, and then running rstan::sampling for each of the 1000 simulations with separate data inputs for each. In order to improve the efficiency of my code, I opted to change the stan model to a vectorized approach, and am feeding in a vector of data inputs. My question is, are the posterior distributions in the vectorized approach independent of one another (from data input to the next).