We address the issue of parallelizing constraint solvers based on local search methods for massively parallel architectures, involving several thousands of CPUs. We present a family of a constraint-based local search algorithms and investigate their performance results on hardwares with several hundreds of processors. The first method is a basic independent multiple-walk algorithm: each processor runs a local search starting from a distinct initial configuration and the first one which will reach a solution will notify the others and stop all computations. These simple methods have good performances, and good speedups can be achieved up to a few hundreds of processors. Then we consider 2 versions with communication between processors: 1) every $c$ iterations, each processor sends the current value (cost) of its configuration to others and a processor who received a better cost from another processor can decide to stop its current search with a probability $p$; 2) the number of iterations corresponding to the cost is also transfered. Both the received cost and the number of iterations have to be better for a processor to decide to draw a probability and restart. Several experiments involving more than 100 processors have been conducted and different values of $p$ have been tried to consider more or less "autistic" processors. However results show that it is very difficult to achieve better performance than the initial method without communication.