That's an interesting question, and one that might make sense in certain use cases. The default assumption is usually that the issuer of the claim would want to keep a copy of the claim. However that is not required. If it is a public claim, the issuer could simply write it to the ledger and not keep a copy. If it is a private claim, the issuer could simply write it to the agent endpoint of the identity owner the claim applies to—and not keep a copy.
So there's no requirement for the issuer to keep a copy. It comes down to the issuer's own business rules.
Although the primary purpose of the Sovrin ledger is to store the essential data necessary for self-sovereign identity—IDs, keys, pointers, and proofs—there is not (yet) any rule in the Sovrin Trust Framework that says identity owners cannot write other public claims to the ledger.
However the issue is one of storage costs. If the Sovrin ledger is going to scale to trillions of identities (and Sovrin Foundation chair @phil has already calculated it will need to do that when you include all the people, organizations, and IoT things on the planet), it's important that each of those identity records be as thin as possible.
So additional data that is not strictly required to establish and protect self-sovereign identities SHOULD be stored at agent endpoints (or other service endpoints) in a DDO. Those endpoints can expose cloud services or other storage options that should be able to handle "everything else".
@phil and others on the Forum: do you agree? This is important, because we may need to capture this policy explicitly in the Sovrin Trust Framework.