Kelly these are sealed combustion forced air furnaces, not unlike what you may have at home, and with many of the same safety features. Do install smoke, CO, and LP detectors in your trailer, but the furnace itself is a very safe design.
The biggest objection in my mind is not safety, but noise, being so close to the bed. We use ours in the evening and morning, but just snuggle in our sleeping bags during the night. Full disclosure: 5°F guarantees that we're at home in front of the wood stove, not out camping!
Another issue is DC power consumption if you camp off-grid. The blower can run down the battery
Beware the catalytic (Wave) heaters, though. They are quiet and do use not power, but they involve combustion inside the cabin, so (1) adequate ventilation is essential, and (2) they can create condensation issues, since water vapor is a major byproduct of combustion (factory furnaces exhaust combustion gasses to the outside).