I think your question may stem from a misconception.
An ordinary Bitcoin address is the hash of a public key (plus a few extra bytes), which in turn is associated to a private key. So in that sense an ordinary address has a private key.
Multisig addresses do not work like that. The address is not the hash of a public key, and so there is no private key directly associated to the address. Rather, a multisig address is the hash of a script. The script in turn lists the public keys whose signatures will be needed to redeem the transaction. I think it is these keys that you are referring to as "existing addresses" in your comment.
So spending the transactions requires you to know the private keys of the intended recipients. There is no other private key involved.
Also, there is no real problem if other people know the script whose hash is the multisig address. The only extra information it contains is the public keys of the recipients. This might help someone identify who those recipients are (though the script has to be revealed in order to spend the transaction, so this information will eventually be public anyway), but it doesn't help them spend it.
For more information, see these answers:
How will multisig addresses work?
What is "Script Hash" address exactly and how does it work?