Privacy is a factor and part of the basis of the argument in Roe but that idea doesn't really hold up if you consider abortion to be a crime. An example in the medical setting is how some injuries have to be reported. It would be reasonable to point out how substance abuse isn't necessary reported but I doubt prolife people like that policy either.
To be clear I am pro-choice and there should be federal protections if not a full blown amendment for abortion, birth control, and other medical procedures but currently those protections are limited. That is how some states think they can do shit like demand medical records from other states. In general the idea of protections for privacy seems to be more limited than it should be.
For this particular issue the obvious thing outside of what should be legal would be that the states have no reason or jurisdiction to know anything. If there was a crime going on in that other state the demanding state has no grounds to charge anyone with anything. No crime was committed in their state even if a crime did happen. Said crime would be the business of the state in which the crime took place.