As a supporter, I can share my experience. I met my veteran online, it wasn't a dating site, neither of us expected to develope any relationship at all. It should have been a one and done conversation.:)
We wrote and skyped for over a year before we met. He let me stay in his home for 2 weeks for that initial meeting. He didn't have any problem taking naps, knowing that I was right there and would have the opportunity to do whatever I wanted while he was asleep. He even asked me to wake him up a couple of times after we had a late day out which, therefore, pushed his nap back and he didn't what to sleep too late and screw up the night. Oh, he also sleeps with his (upstairs) bedroom door open, says he's slept in too many tiny, closed off rooms. He gave me a house key, so I could come and go as I chose. We kept our electronics and wallets out and accessible most of the time. I had tried to offer him £400 ($600) for letting me stay with him, but he refused. So there were many, many times either one of us could have snooped or stolen from the other. We cooked together, which involves knives, of course. He didn't mind me squeezing by him in tight spaces or standing behind him when I had to. In public, he was more concerned with just knowing where I was so he could keep me safe if something happened.
Hope that helps. I'm sure it's different for everyone. I would say, yes, his home is his safe place, not because he is afraid elsewhere, but simply because hypervigilance is exhausting and his home provides him with a quiet, calm, relaxing place to destress and unwind. I think he showed a great amount of trust to allow me into his space. I'm honored.
To address a bit more of your angle of being in someone's house without them there...I don't have PTSD, but I wouldn't let someone stay in my house without me if I didn't trust them to respect my property. I can't imagine anyone would.