What you’re describing is just email.

Yes, you can host an email server.

Maybe what you’re asking is if this can be done within a private network, so you wouldn’t need to buy a domain and set up dns and all. The answer is again yes, but it wouldn’t be much good. Either both servers are on the same LAN, or they each have a static IP that is known and trusted by the other. In both cases there are much better ways to communicate than email.