The bit about public/private keys being for encryption/signing respectively is not a general fact of public-key encryption: it's specific to RSA, where public and private keys have (at least in the textbook) the same form. Switch to Diffie-Hellman and/or ECC based systems and the two kinds of key look completely different, so you cannot encrypt with a private key.
The short answer to your question is that if Person1 has the private key of Person2 then something has gone badly wrong already, and most likely no-one will get any security at all. This kind of thing happens all the time (I think github has a special error message for people who paste a private key in the field where a SSH public key is expected to setup SSH keyed authentication).
The full answer: if Person1 uses their own private key, if they're doing textbook RSA then anyone with Person1's public key can recover the message. And public keys are, well, supposed to be public. In the real world, RSA keys and other data structures are a few steps removed from the textbook - and a sensible crypto library should not offer the option to encrypt with a private key in the first place. You would also typically have separate subkeys for encryption and signing, which again would be stored in different formats (it's a very bad idea to use the same key for both, even if you could in theory).
In textbook RSA, you have a public modulus N, a public exponent e and a private exponent d. In encryption mode, you compute a ciphertext c from a message m as c = m^e (mod N); to decrypt you compute m = c^d (mod N). To sign m, you publish s = H(m)^d (mod N) and to verify a signature s on m you check that H(m) = s^e (mod N). So the public and secret operations are interchangeable. In fact it doesn't matter which one you call e and which one you call d, the two are interchangeable : x^e^d (mod N) = x^d^e (mod N) = x (mod N) for all 0 < x < N. (Well, almost all - as long as you don't hit a factor of N. In which case you're toast anyway.)