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If an account had ETH before the DAO-fork does the ETH now not exist on both the ETH and the ETC chains?

A Poloniex notice says "All users who had an Ethereum balance at the moment of the fork now have a matching balance of ETC". It seems that if the ETC chain lives, then account holders can spend and trade on two chains. Effectively, is it possible to transfer your ETH balance to someone else on the ETC chain, and have it remain in your ETH account on the main ETH chain?

We also have a couple of posts referencing this:

https://steemit.com/ethereum/@pauls/ethereum-fork-step-by-step-guide-to-safely-splitting-your-eth-etc

https://steemit.com/crypto-news/@phistr/ayvsf-ethereum-step-by-step-guide-to-protect-your-eth-and-etc-funds-and-avoid-replay-attacks

A quote:

Step 2: Send funds to Poloniex. Generate an ETH deposit address or use an existing one. Then send a test transaction (for example 1ETH) to this address. You should be credited ETH and the same amount of ETC. If so go ahead and send the rest of the ETH.

Can someone please help explain whether a double spending issue exists?

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Yes, an account before the DAO-fork will have Ether on both chains: ETH and ETHC.

It's not a double spend in the usual sense, because when you send ETH to Alice, you cannot send the same ETH to Bob. You could send ETH to Alice, and ETHC to Bob, but they are separate blockchains (similar to how you could send BTC to Charlie, except that ETHC account balances are those of ETH accounts before the fork).

Replay attacks are different from double spending. Without taking precautions, when you send ETH to Alice, she could replay that transaction and get the same amount of ETHC from your account: if she does this, you may be upset that you no longer have the ETHC to send to Bob. Likewise, if you send any ETHC to Bob without taking precautions, Bob could replay that transaction to get your ETH.

EDIT Question added: "is it possible to transfer your ETH balance to someone else on the ETC chain, and have it remain in your ETH account on the main ETH chain?"

No, you need to do something to prevent replay attacks, for example: How to prevent a replay attack between two competing chains? After you take precautions, if it involved moving the ETH to another account, you can send it back (but it is probably safer to not send it back). It is not possible to send a single, simple ETC transaction and expect to be safe from a replay attack.

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