Validating nodes need to download and verify the signatures. This is true now, and will be true after SegWit.

Validating nodes do not need to keep signatures around after validation. This is true now (pruning), and will be true after SegWit.

Anyone who has a transaction included in the chain can construct a proof that this happened (SPV proof), and this proof includes the signature that was used. This is true now, and will be true after SegWit (using the wtxid merkle tree instead of the txid merkle tree).