Where is the SSH RSA modulus actually sent? This might be a stupid question but: When RSA key pairs are generated, there is a pubkey and a private key file. Where is the modulus? Is it appended to the keys? I can't find any information about that.
The modulus is part of the public key. For RSA, the public key consists of the pair $(n,e)$ where $n$ is the modulus and $e$ is the public exponent.

The SSH public key file contains both, typically one after the other. There's a number of ways to encode this information. The gory details of these encodings I leave to this in-depth answer (it includes a section on SSH).

The SSH private key file contains pretty much all numbers associated with the key generation process. If you want to take a look yourself, try running the following command

openssl rsa -in test.key -text


where test.key is your private key file.