Thanks for coming back and posting a few of the reasons.
Personally, while I think it is okay if that is how you feel for you to do, I think I DO know teens pretty well and I know mine enough to notice if things seem "off" at which point I might then for that reason, keep closer tabs.
Since we do not have texting on the phones (the kids do not want it

) and both kids usually leave their phones downstairs on their chargers anyway, I don;t worry about sexting.
watching porn: this is values thing and everyone has different comfort levels. To ME occasioanlly checking out the now "milder" stuff is probably developmentally appropriate at some point and also nothing I need to know about

We have hammered into the kids' heads that there are some sights that are illegal to even look at and many more that are beyond anything normal or safe and many that will put a virus on their computers so fast they won't know what hit them, so don;t go messing around that stuff online beyond the tame things you can find on youtube without following outside links. If they need to see more, they can avial thmeselvs of the "standard" television channels after 10:00 pm any night DH and I are out here in Germany ans just not tell me about it

I totally get that others do not want any porn of any level seen by any teen (or anyone) in their house. I think that is fine. I think not having computers in bedrooms might be one way to fix that issue and still close doors.
drugs--if there is a drug problem it will be there whether or not they can do them in their room. I hope i notice the redflags and find out about it. Actually, I hope IF there is a serious issue like that they DO try it at home where I am much more likely to stumble across it and learn what is going on and then be able to get them help. Again, I understand that my viewpoint doesn't work for everyone--but it works for me and mine and I thought I would explain what I am thinking and why letting doors close doesn't mean I don't know my teens or teens in general.
facebook--I am friends with my kids and with most of their friends and would see it (yes, I know they can v\block things). Besides, the whole not having a computer in the room would keep it from occurring in the room it if I thought that was such an issue. And really, they can post to facebook in about 60 seconds--I don't have my eyes on them all the time even when they are in the seam room as me anyway so it could happen with a door open just as easily.
cutting--When I was a librarian, we had three teens we often found cutting there. Kids cut at school. They cut at home in the bathroom (or do bathroom doors have to stay open too), etc. If this is an issue, a parent's concern should really not be stopping the behaviour, it should be helping their child deal with whatever issues are driving them to that behaviour.
Blocking out the world: heck, sometimes
I do that! I didn't know it was a bad thing to take a break. In fact, I do not feel it is. I am guessing you mean as in the teen never participates in "real world" things. That would be an extreme case and I think many posters (myself included) would say that is different and something that would need to be dealt with IF it occurred, but not something to anticipate. Again, this is an issue where i would want to find out WHY my child felt the need to do so and get him/her help.
Sex? You saw that the OP was about being ALONE in one's room, right

I guess you could be thinking ************ here? If so, personally I do not think that is immoral AND I think most kids who wanted to do that and could not close the bedroom door would likely start taking really long showers or some such.
When you say NO DOORS, does that include YOUR bedroom door? What about bathrooms? Will you allow your teen to shower in a closed bathroom? Or use the toilet? What if the only adult home is your DH? Is she required to let him check on her in these situations, or do you really only mean kids' bedrooms?