I guess I'm lucky enough that both backstages in my school are connected to a hallway that I can use to put my prop table if need be. I'd suggest though making the actors check every night to see if their
props are there anyways. I've found that, although they may not care about other people's props, they do care if theirs are not there for fear it may mess up their scene. If I don't get a props manager for a show I usually delegate that responsibility to a member of my backstage running crew. That way I'm not always going back there myself to check on them as well plus at the end of the performance that night I know that there's someone who I trust hunting down the missing props that never made it back offstage or disappeared right before they went onstage.
I'd definatly shelve if there's really no space though.