I'm a former EMT. I've worked in both a rural service and an urban one. In the rural service, we rarely had police support for medical calls (as opposed to trauma calls). If someone was ODing, and violent, we called back and got police support, as legally we are not allowed to restrain. In the urban service, anything that came in as 'altered mental status' got police support, quite often an officer partnered with a new policeman or woman. It ups their experience in a relatively low risk situation. Oh, and we often bagged up any prescription drugs, plus any other 'interesting' things we found. Takes less time just taking it all to the ER than writing down all the Rxs and the doctors need to know all the legal and illegal drugs they are taking, for interactions and overdose symptoms.
As a side note, the scariest mislabelled call I think I ever went on was to a "granny down, no rush" (older lady fallen, non life threatening) that turned out to be an older lady who had sliced open her ulnar artery while chopping carrots for supper. Scared the pants off me when I unwrapped her wrist to look at the 'little' cut she had given herself before she started feeling woozy. Keep in mind how rare it is to actually know what you're getting when you go to a call, even if the person sounds like they might have a clue. Going to an OD, especially if the cops haven't arrived yet, your EMT's might not even enter, if their protocol says to wait for police support. If they do go in, you can bet they will be checking out the surroundings for potential threats first, then looking at your character. They might be subtle about it, but it will be done.