Many moons ago, I had a polite interaction with a deputy sheriff in the courthouse about how I wasn't comfortable with having the patient not be transported.  He had some weird disease that I had never heard of, and their rule was, if it was a chronic condition, it should be treated by the prison med staff, but if it was an acute issue, he should go to the ER.  got escalated to the Sgt, who called the Lt, who called the captain, who called someone else, who finally agreed to let us take him to the hospital (not the one he requested, but the one they were contracted to used). 
I've made mistakes in my past and not advocated strongly enough to LE that we should transport a patient, but most of the time, if the patient is sick enough, they will let you transport them.  but if the patient is in custody (IE, has the silver bracelets applied), the officer is responsible for what happens.  While I'd had my conflicts run-ins with cops whose egos were larger than mine, I'd say that's the vast minority of the interactions.  And while I've dealt with more than my share of patients suffering from incarceritis, very few were legitimately sick enough for me to argue with the officer that they needed an immediate transport to an ER.