Yes both LSR could advertise given FEC, say 10.0.0.1/32 with label 10 to each other.
Then if IGP says to LSR1 10.0.0.1/32 egress interface is towards LSR2, it'll impose (or swap to) label 10 and send towards LSR2. LSR2 then will find egress interface being something else than towards LSR1 and swap label to what ever that direction has advertised, might be still label 10, or might be something else, does not really matter at all.
Labels are completely local today and some RFCs dictate that is how it should be. Personally I'd like IGP labels to be global for simplicity. Because MPLS LSR does not know how labels look from anyone else's POV, we need hacks like tLDP (targeted LDP) when implementing rLFA (remote loop free alternative). We need the tLDP to learn bindings of remote node.
With regards label scope, label space today is chassis-wide in every device I've ever seen, but standards fully allow per-interface label spaces.