The entire point of partitions is that not all full nodes are validating (verifying) all transactions.
Thus of course the full node that wins a block (in PoW, and analogously ditto in PoS or consensus-by-betting) is trusting the validators of other partitions to not lie to him.
If that full node had to validate every transaction in every partition, then there wouldn't be partitions any more. The entire reason to make partitions is because verification costs are too high when every full node has to verify every transaction. Partitions exist to aid scaling.
Can we be clear on what you mean by validation? Validating a transaction (i.e. checking it is protocol valid) has no PoW cost associated with it, any full node can do this. Therefore any full node can reject an invalid transaction before it gets propagated around the network.
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.