(I realize this picture isn't the greatest to show what I mean, but I was thinking about this in the moment and forgot to get a better screenshot of me playing the actual game vs this screen here)
Mostly as someone who has been streaming the game here and there, I turned on the "hide names" option to be careful about potentially uncomfy names being shown.
I know there has been some talk of a party finder mechanic in future updates/full release but there is no way to hide your party member's names. Which you can create an overlay to hide the names above your health bar and stuff but the icons showing where your party members are on the map in relation to you shows an icon AND their names.
Not a big deal for the "right now" with only being able to make parties based on sharing a code with people, but if there will be a random party finder mechanic in the future, having these names up could be a problem for people who stream, where we could otherwise just have the little icon on the screen to point out the direction members are in.