Weakness in using only one RSA key pair for two-way communication?
The question and comments seem to be asking the following:  If an implementation of RSA is used in the following way, is it still secure?  An RSA modulus $N = pq$ and exponent $e$ are generated, and (N,e) is given to Party $A$ and $(p,q,e)$ is given to Party $B$.  Then, the parties encrypt their communication where Party $A$ encrypts using the modulus and exponent, while party $B$ encrypts using the private primes and the exponent.

To really answer the question, we'd have to define what encryption padding method is being used (OAEP, PKCSv1.5, etc) and what notion of "secure" we want.  We'd also have to define what it means to encrypt with the secret key instead of the public key.  But as long as the encryption methods with either key are generating the same ciphertext (more precisely, the same distribution over ciphertexts), then whatever reasonable security notion we want will be achieved in this setting under the same hardness assumptions.  This is because the adversary is seeing ciphertexts that are all indistinguishable for ciphertexts generated in the usual way, so the basic security achieved will transfer over easily in provable way.