A wallet stores private keys, not bitcoins (and a watching-only wallet just stores addresses, without their private keys). Your bitcoins are “stored”, for want of a better word, on the global distributed ledger. You can verify this by looking up your address/addresses (not your private keys!!!) on a blockexplorer web site such as blockchain.info — this will show you that your bitcoins are at whatever address/addresses you sent them to. You can do this even if Armory hasn't downloaded the blockchain.

In order to construct a transaction, Armory needs a local copy of the blockchain and it needs the [public] address/addresses from which you're spending. But the transaction will not be valid until you have signed it, and for that, you need the private keys. In a secure set-up (cold storage), you construct transactions in a watching-only wallet (connected to the internet), and you sign them in a full wallet (on an air-gapped computer); such that your private keys are never exposed to the internet.