I have two separate circles of friends who camp or have bring your chair lawn parties. In both sets everyone brings chairs (enough for themselves and maybe even a extra if they have one). The chairs are set around the firepit or the lawn and they are considered communal for the entire duration of the event, anyone can sit in any chair at anytime, because there are always enough chairs for everyone, so specifics aren't really considered. Especially since chairs do get moved, and many of these folding chairs look almost exactly the same (especially once it gets dark and/or the beer comes out).
If someone had a specific chair they brought and they needed to sit in, it wouldn't be a problem for them to communicate that, but under no circumstances would it be some automatic thing where people automatically knew to get up and give someone back a chair.
People sit in various chairs that aren't necessarily their own for all sorts of reasons - to sit near someone they are having a conversation with, because half the group is off doing something else and the few people left want to sit together, to move out of the line of smoke from the fire, to move out of or into the shade/sun, because it was the first empty chair they came to, etc.