Recently the Japanese Splatoon Twitter (translated by Siliconera) account revealed a bit about how the Nintendo voice chat system will work on the Nintendo Switch. The information came in the form of two tweets briefly describing how the system would work for Splatoon 2. There was also an image included with one of the tweets showing a squid using its phone at the same time as the Switch.
It seems that, at least for Splatoon 2, the voice chat and online lobby system will be done independent of the game itself, through the Nintendo Switch smartphone app. The squid character has its phone sitting next to it while it's playing the game, this seems to suggest that you will need your phone the entire time while playing in order to use voice chat.
Roughly translated the tweets state that the system works in conjunction with the game allowing you to meet up with friends in lobbies, invite friends using social media, and chat with them in game. The tweets also state that in a private match you can talk to everyone in the game before it begins and then only your team during the game. Interestingly, it adds that you can not talk to someone you don't know; this could mean simply that you can't talk to strangers while in a party, but it could also mean that there will be no voice chat allowed outside of parties, we'll need to wait for more concrete information to be sure.