My understanding is that spending a coin on one chain could expose the private key of the other coin on the other chain, compromising its security. If this is true, how is this prevented or mitigated?

No, that’s not the case. The main issue is that any transaction on one chain can be replayed on the other, if they have compatible rules as in this possible split between Bitcoin and RDTS.

If there is a persistent chain split, people will need to separate their funds on the two chaintips. This could be for example achieved by sending a transaction that is only valid on one of the two chains (e.g., by violating one of the new rules RDTS introduces), or by simultaneously sending transactions to yourself on both chains at the same time with different recipient addresses, hoping that they both go through. If the feerates are sufficiently different on the two chaintips, you can send a transaction on the chain with the lower feerates first, and then spend the same UTXOs to a different address at a higher feerate on the other chain. Alternatively, you could wait for some coinbase outputs to mature and acquire a UTXO that descends from a coinbase output that only exists in one chaintip.

Once you have any UTXO that only exists on any one chain, you can use it to split all of your funds by spending that UTXO along with others from your wallet. All of the outputs of those transactions will only exist on that chaintip. Using such UTXOs to “vaccinate” your transactions against replay attacks, you can then reliably spend and/or sell funds on either chaintip.