I know I am always really careful with other people's thigs, especially valuables. If a person tells me "be careful with it" or "don't break it" of course I'm going to try to do what they asked of me. Then again, I am a loyal person who tries to obey. However, if I don't like the person, I may do the exact opposite and try to ruin the item the first chance I get.
As a libertarian this is an argument I would generally make in favour of private property rights. People will ALWAYS want to take care of their own property. Taking care of shared (public) or someone else's private property depends entirely on respect/fear. If you have adequate respect or fear you will take care of it but in general people will always be more careful with their own personal private property.
I would assume they would care less about someone else's valuables. It is not their valuable so why would they protect it or treasure it like their own? People are protective with what they own. But they care less about what others own. Here is a common example of carelessness where someone drives down a street and hits a parked car and keeps driving. That person knows if it was their parked car they would want to know who hit it and require them to pay for it. But they don't care because it is not their car.