Would data be secure if a cryptographically secure PRNG was used for encryption?
A PRNG with a seed $S$ whose output is combined with the plaintext is called a stream cipher with the key $S$. So assuming that the details are filled in correctly, what you're describing is a stream cipher algorithm with two session keys, one used in each direction. Each seed/key must be unique, and the combination must ensure that each output bit has a $1/2$ probability of being flipped (xor is common, but addition of bytes would work too).

If you have an encrypted communication channel, then it's already using a stream cipher, which may be built on AES. (Note that AES is not a stream cipher — it's a family (128, 192, 256) of pairs (encrypt, decrypt) of algorithms that operate on blocks of exactly 128 bits. There are ciphers built on top of AES with a mode of operation, which result in a block cipher or stream cipher.) For example, AES-CTR is a popular stream cipher. AES-CTR is also a popular PRNG, for exactly the same reason.

Combining your scheme with another cipher would mean, in fact, chaining two ciphers. This is hardly ever useful, but is at least as strong as the weakest cipher if the ciphers are not correlated. (Possibly not as strong as the strongest cipher, because the weakest cipher may leak information.) Correlation can lead to catastrophic failures — for example, if you encrypt twice with the same stream cipher with the same key, you get the original stream.

A stream cipher only ensures confidentiality, not integrity or authenticity. For example, AES-CTR is trivial to modify: flipping a bit of ciphertext results in the same flipped bit in the plaintext (this applies to any stream cipher that uses xor as the combination function). HMAC or some other form of MAC can ensure integrity and authenticity. Alternatively, you may use a mode of operation that performs authenticated encryption, such as CCM or EAX which are CTR combined with CBC-MAC or CMAC.

Further reading: Is modern encryption needlessly complicated?

 I'm only rehashing what's been said in comments really, just so that this question is answered.