Rings are difficult, especially if like my mother, the person won't part with them, but then takes to hiding them away. Her rings did go missing, and never turned up. I can't say categorically that nobody nicked them, but when a person starts wrapping things in tissues, or knickers (even dirty ones, I'm afraid to say) and starts hiding them all over the place, it's easy for things to be inadvertently thrown out. Especially tiny things.
Looking back, I wish I'd thought to tell her I was taking them away to 'have them valued' but she would have hated parting with them and in the early days at the CH it was hard enough anyway to keep her reasonably un-agitated. Might add that a valuable ring of an aunt's went missing in her CH, and it happened to coincide with my sister and BiL visiting. After an 'absolutely everywhere' search, it turned up in her waste bin - wrapped in a pair of decidedly manky knickers...
But I would certainly label absolutely everything, photos, nick nacks, the lot. An indelible marker pen is probably good for labelling anything like ornaments on the underside. Otherwise it's almost impossible for staff to know which room to return 'squirrelled' items to. A large old framed family photo once appeared in my mother's room - she had almost certainly helped herself - it was unlabelled and none of the staff could be sure whose it was. The only way was to leave it in the kitchen area, which most visitors used, in the hope that a relative would recognise it.
One minor thing - I bought my mother a new dressing gown a couple of Christmases ago and in addition to a firmly sewn-on name tape, I stitched the belt very securely to the gown. So that unlike its predecessor, the belt never went missing, never to be seen again.