I think you're on the right track with revising the function names as you've suggested. That will allow you to clarify the intent of what is happening as well as minimizing code duplication.
I would suggest calling processIntersections something different though. updateIntersections or checkAndUpdateIntersections would be two suggestions. The intent is to get the function name closer to what is actually occurring.
I'm not sure I understand the concern over 'const-correctness'. You said that workerIntersections would be a private method, so it's effectively hidden outside of that class.