No, it doesn't. Third party keyboards have two levels of access and they come pre-installed with the much more limited one (no access to internet whatsoever, no matter if you're on WiFi or 3G/4G).

Doesn't stop Google's keyboard to periodically pester you to give it full access "to get the full experience" (or something along the lines) but if you don't, it will never get or send a single byte to Google's servers.

Trust me, I researched. I too didn't want the huge irony you mentioned to hit me. And it doesn't.