good answers in this thread so far. My experience, if y'all don't mind, concurs pretty close to ridge runners answer. A previous home in the country where we had chickens, I built a coop and run I was real confident in it's predator proof protection for my girls. Having lived there for over a decade, I wasn't the least bit apprehensive to leave the pop doors open all the time.
We moved into the city for a few years, now we're back in the country. I've built another coop/run and I am not the least bit worried about it either, BUT I've only lived here a few months and just not completely comfortable with leaving the pop doors open all the time yet. I have a couple of trail cameras I plan to set up on it so I can see what, if anything, comes around at night. Also, I'm not completely finished with the run portion, I still have to do an apron and secure from 'diggers'.
Once I know what I have/don't have to worry about, then I'll consider whether or not I'll leave the pop doors up all the time.
So it's all personal preference and confidence in what you have to worry about...........or not