c++ generate thread safe random numbers

I've read some similar questions to the one I'm asking but the answers don't seem complete or completely clear to me.

I'm trying to parallelize a parameter scan that requires the repeated generation of a set of random numbers. With only one thread I currently do something like this:

int main() {
  //Get random number generators
  typedef std::mt19937 MyRNG;
  std::random_device rd;

  //seed generator
  MyRNG rng;
  rng.seed(rd());

  //make my uniform distributions for each parameter
  std::uniform_real_distribution<> param1(-1,1);
  std::uniform_real_distribution<> param2(-1,1);

  double x,y;
  //Do my scan
  for (int i = 0; i < N; i++) {
    x = param1(rng)
    y = param2(rng)

   //Do things with x and y*
  }

In this way I get a new x and y for every scan. Now I want to utilize multiple cores to do this in parallel. So I turn define a function void scan() which essentially has the same contents as my main function. I then create multiple threads and each have them run scan(). But I'm not sure if this is thread safe using std::thread. Will my random number generation in each thread as it currently is be independent? Can I save myself time by creating my RNGs outside of my void function? Thanks.