Ah, another example of where I am a social pariah.
When this issue came up with my children, I noticed something. I'm a noticy kind of person. I noticed that *I* don't share with other people. And I'm not unusual.
We do not live in a society where it is 'okay' to decide to share someone else's stuff without their permission. We have a big thing about ownership and property rights.
But we don't raise our children in that truth. We don't share with them, and we don't really expect them to grow up to share any more than we do. In fact, in a couple of years you'll be encountering parents who are very upset about their children sharing clothing, personal items and family resources with other kids, although it is exactly what they were taught as toddlers.
We never made our kids share anything. In fact, they were always at liberty to shut the doors to their rooms and bar anyone using anything they owned. Just as I do. I do not share my makeup, and friends are not at liberty to put on my clothes just because they are in my home. Yet, this is very much what you are describing wanting your 2yo to do.
Strangely (or perhaps not) I have children who are extraordinarily generous, with each other and other people. They will happily share what is theirs almost every time they're asked. Of course, they grew up knowing that their stuff was actually theirs to control -- they knew they really owned it, so they don't feel like their stuff is in anyone else's control (even mine), so they don't feel any need to be overly-controlling of what happens with their stuff, just to prove to themselves who owns it.
I considered it my job to respect my children's wishes regarding the use and care of their stuff. Particularly when they were very young and more easily pushed around by others.