The experiences will vary from player to player, and depend on the immortal as well. I can only relate my own experiences. Generally, I prefer to do "non-corrective" NPC actions, to switch into the bartender and gossip with a patron, or the fortune teller in the bazaar in order to utter dire prophecies. Unfortunately, that's not always possible.
Circumstances under which I might possess an NPC (in no particular order):
Please note that there's a hierarchy on the staff -- not all immortals can communicate directly with players, and some (the majority, in fact) need to find IC ways to answer wishes, such as possessing an NPC.
There are certain player reactions to realizing there's an immortal in the room, most of which tend to contribute to a somewhat jaded attitude on the staff. A few of these:
Personally, when I play, I play, and don't really care too much who else is around. My character wanders through the city, looking for clove scented incense, or rides around talking philosophy with her kank, or whatever. Most of the NPC bard songs are ones I wrote when running a bard, and while some of them had their first moments off the cuff in the Trader's, others I sat and roleplayed out making up, muttering as I tried to figure out the rhymes. Why? Because I tend to like my characters. They interest me, and I try to flesh them out psychologically as much as possible. I like RP-ing with other people, and I'll choose that over sitting alone -- but generally my characters have plenty to keep them occupied even when there's no one else in the city.
However, I know that's not everyone's style, and that's cool too. I do think that when you're playing, you need to take into account the virtual world around you (I think I've posted about this before, so I'll avoid rehashing that) - and think about whether what your character is doing is reasonable or not.