I suppose for me it depends on the group content and who I'm doing it with.
I prefer to run group content with people I know as I do have social anxiety (especially with DF, even though I have had NO bad experiences in the 2+ years I've been playing - yep, tried telling my brain that but it just pops up like the Microsoft Paperclip saying "Ooooh, looks like you're about to heal a DF Dungeon with some randos, would you like some anxiety-inducing thoughts while you play?"). That said, I feel more relaxed in Alliance Raid/Normal Raid roulettes (usually because you have a backup healer to assist if things go pear-shaped) and will happily join them. I'll also do Frontlines because it's a bit of a laugh and if you die, well, you still get exp so it's not the end of the world. Dungeons and Experts Dungeons I'm a little more apprehensive with, and especially first time I prefer duty support/trust to get a feel for it so I don't waste people's time screwing up over and over again.
I do play on US servers whilst based in the UK so suffer some lag which has an impact in group content (usually see mechanic telegraphs about 1-2 seconds after more local players so get hit by them if I don't just follow the group the moment they start moving) and it does cause some clipping in stricter rotations (one reason why I love playing Healers - no strict rotation to follow). I also do struggle sometimes trying to remember rotations or keeping track of everything (dots and positional) so I'd never do Savage content but I'm alright with that as it doesn't really interest me.