In general terms -
Locality is preserved in one of the two ways -
The two entangled particles start with a complex/capable enough wave function that generates the correlation even when independently (without need of any mysterious link) working on two particles. - This is mainstream, most accepted quantum explanation. I guess this is referred to as superposition principle, multiple particle joint amplitude .. If someone has learnt the QM math well enough, then there are high chances that this will be the best solution in his/her opinion.
Many people still seem to think that all possible classical explanations of correlation have not been exhausted, and there can be some explanation that still needs to be discovered.
The correlation means following behavior -
a) anti correlation in every direction, always
b) statistically 50/50 in any one direction for each particle independently
c) statistically Sq(sin(A/2)) correlation at relative angle of A.
d) Please comment if any additions/correction are needed to the above three.
Local variables/plans (i.e. fully pre-determined plans) are not sufficient to exhibit the correlations and this has been mathematically proved by Bell's inequality.
Bell's inequality disproved the ability of "local" "static" "variables" to describe the correlation. It did not disprove same for "local" "functions" like superposition etc.