Does merged mining require clients to download both blockchains? Suppose I want to start an AltCoin, and set it up for merged mining with Bitcoin. Will it require the AltCoin end-users to also download the Bitcoin blockchain to verify the hashes embedded in the blocks, or do only the miners need a copy of both chains?
Those running a miner against a pool don't.  The pool operator would need both.

As stated in the comments below, normal users of the alternate system will not need the bitcoin chain but will need the alternate chain, and normal users of bitcoin will need the bitcoin chain but not the alternate chain. Once the block is mined, nobody cares what other chains may have been involved.