It depends. If you import whole wallet.dat into altcoin wallet, you share both present and future (next 100 or so) private keys with the software.

If you transfer your bitcoins to new address provided by existing bitcoin wallet, chances are the altcoin wallet also has a private key to those addresses.

To be safe, you'd have to transfer existing bitcoin funds to completely new wallet. Also, for privacy reasons you would need to make as many transactions as UTXO you have, which means paying a hefty fee tens or hundreds of time depending how much you use your wallet.

Also note that transactions that consume multiple UTXOs on one chain lead to privacy leaks on other chains.