It's hard to describe... at first it feels almost like an erection, only much much smaller and with all the "tension" concentrated in one tiny area. (Several cis women have said the same thing to me, actually.) As far as I can tell, my clit does get firmer/erect, but it's not possible to see that with the naked eye. With time and enthusiasm, yes, my labia become a bit engorged and vastly more sensitive, although the sensation is probably because by then I'm focusing on the area! I can *always* feel touch/sensation in both sets of labia, but when I'm into it enough I about come out of my skin over even gentle stroking there.
I do get wet, although not in copious amounts until near the end.
I've only ever felt a dilator, so that's *not* an arousing context; it feels like, well, shoving a hard plastic object in there. I'm also primarily into women; dunno what to say about "desire to be penetrated," because to me that's purely mental and depends on what kind of sex you want. I think it'd be cool, I guess, but it's not a driving need.