Validating (a.k.a. verifying) also means checking that it isn't a double-spend, that the funds exist (either via UXTO or account balance).

In a partitioned design, only the full nodes (a.k.a. validators) for each partition would validate and propagate the transactions for that partition. So yes you are correct to imply that partitioning means the P2P network is partitioned also (because otherwise DDoS spam amplication attacks would be plausible if peers relay that which they do not verify).

I think all that should have been clear just by thinking about the only way partitioning can work. I am just wondering why you can't deduce these sort of things and instead need to ask?

Note that validators can be computing a PoW block based on a hash of their partition and a hash of all the other partitions. Don't forget the power of Merkel trees.