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We have recently hiked in Slovakian mountains where bears live. I’m not too concerned about the bears during the day, as we make enough noise to avoid an encounter. But I am not sure about sleeping there.
We slept in bivy sacks in the open, I hung our food on a tree somewhat distant from our sleeping place and generally followed all the usual suggested precautions. But when we could hear the bear grumbling in the woods in the night, we were pretty scared anyway. I feared that the bear might get quite close without us noticing, and I’d hate that.
How safe is sleeping in the open in a bear country, then? Is the general idea that the bear will simply not notice? That the area is large enough to make the encounter improbable unless we do something wrong, like cook in the camp?
With the exceptions of grizzly bears and polar bears, bears are nuisance animals that are after your food. They have no interest in you except as a source of food or as an obstacle to obtaining your food. In specific, extremely popular camping areas (e.g., Yosemite Valley), bears become habituated to people and are much more common, so they are more of a nuisance. The farther you get away from popular areas, the fewer bears you will see and the less likely they'll be to be habituated to humans. Sleeping in the open is no better or worse than sleeping in a tent; a tent is not a barrier to a bear. If you don't want to be bothered by bear noise at night, store your food far away from where you're sleeping. If you use a bear canister rather than tying your food in a tree, then the bear may be more likely to give up quickly rather than keeping you awake all night. I usually put my bear canister about 30 meters from where I sleep.
In Romania we also have a high population of bears, and recently they wander also in forests on lower sea level altitudes (such as ones at 300 m); they even enter the towns and villages, picking food from the trash. Attacks have been reported usually in cases when people went too close to bear mothers with cubs.
I would consider the following steps when wildcamping in bear areas:
Pull the food up on a tree ~200 m away from the place where I sleep.
Stay away from spots which might be interesting for the bears (although you can never know what is in a bear's head, there is some probability that he would visit the nice little spot near the creek or spring just to drink some water in the morning - true story, I saw their footprints in the mud).
If there are several people, night guarding shifts could be made, along with a watch fire constantly kept alive (10+ years of scout camp experience stays for this, best effective when there are stray dogs or shepherd dogs in the area - common issue in Romanian mountains; it is so effective that until this day no bear visited our camps :) just the creek 300 m away... that was scary).
I have backpacked and camped in the Rocky Mountains most of my life (and also more recently in the Sierra Nevadas) and have never had any issues with bears. I would say that as long as you're not sleeping with food in your tent, it's highly unlikely you'll ever have a bear encounter while you are snoozing.
As stated in previous answers, with proper campsite management like storing food at a good distance from your tent you will be safe unless you run into a scenario of a mother with cubs and that's rare. I'll add you will be much safer sleeping in bear country than in town where the most dangerous of all predators roam, other humans.