I always leave it there but maybe put it high up on a wall or on a fence post or something if it is a glove/hat cos I know when DS or I have lost things I've always retraced the route to see if I could find them. Once some nice person had picked up DS glove and put it on the wall for us so I've always done similar since.
Do people ever take them home and put an advert in local shops or the newspaper or anything?
Whenever we've lost something I've never even thought until now to see if anyone's placed a found advert anywhere.