A true "alternative internet" would, at this point, require its own infrastructure due to the routers and switches of the Internet using the TCP/IP stack. (assuming that an alternative internet would be a variation on the IP protocol).
The entire issue with the government surveillance of today is that they're (allegedly) storing all data, even the encrypted data, and holding it to either brute-force crack, waiting until whichever cipher/protocol gets broken, or waiting until computing power is abundant enough to crack the encryption-du-jour in a reasonable amount of time.
This is why ideas like Raspberry Pis on balloons have been proposed as true "alternative" internets. Not to get all tin-foily, but I would assume that any undertaking that ambitious would have government agents working from the inside.
If you're transmitting any data that could be illegal, your safest bet is to use some kind of encryption that would take longer than the statute of limitations for whichever crime you're committing to break (you'd probably also have to factor Moore's law into your calculation and pray that quantum computing doesn't become viable in the next x years).