Google will not release the chat logs publically, it's up to the court how to handle them, but the bar for "the public cannot see this" is generally much higher than "well that's embarrassing". If this goes to trial, they will most likely become available.

Remember that the idea of the court is to be public and transparent, with judgement coming from the jury, but also to be judged by society on the whole. So if you're gonna sue your kink provider, be prepared for everyone to know how you get off, because after all, the court is owned by, and serves, the public.