BPMN 2 public and private process

I am trying to model the public process using camunda modeler, see: https://www.omg.org/spec/BPMN/2.0/PDF.
BPMN: Private vs Public Processes – Visual Paradigm BPMN

I want to have separate shared interactions in the second pool.
Here is the the bpmn file where I am trying to test this. Can both pools be executed in parallel when the message is triggered?io.bpmn (6.6 KB)

It is better using a parallel gateway if you want both pools be executed in parallel.