ConcurrentModificationException - will this happen when we have more than one process writing the same table? Do you think we could have the same issue if two processes writing to the same table but different partitions?

it can happen yeah but less likely. both writers might try to commit to the log at the same time

If the two writes are explicitly partition disjoint, they won't conflict at the file level. Since it is an opportunistic commit protocol, if there are two simultaneous commits, one will fail but can be immediately rebased and tried if there are no file conflicts.