How does friendly fire work?
Solution 1:
Allies to Players
I can confirm that I've been shot by my own possessed Patriot before. Enemies were in front of us, the Patriot was behind, and they were stunned with Shock Jockey. Then, I started to take a bunch of damage. So, from what I've seen, you do take damage from your allies. This is most prevalent when you're talking about any AOE effect enemies, like enemies with RPGs.
Players to Allies
The first turret I was able to possess, I unloaded on with a pistol. It wasn't taking damage from anything else, since all the enemies were dead. I unloaded three magazines into the thing, more then enough to kill it five times over. It's health bar didn't move. I've done this to Patriots as well with similar results. I think this is a safe-guard to exploit the Possession vigor from being a one-stop kill for Patriots and the like. This way, your stray bullet's don't 'break' the possession when you don't want them to, but you can't use it to kill anything outright.
Enemies at Each Other
There are times, later in the game (without revealing spoilers) where you might have multiple types of enemies fighting each other. When this happens, they will fight each other and do normal damage. Hence, I've seen one faction having a Patriot and gunning others down. If you get involved, the side you starting shooting at will take interest in you and focus you instead.